This question has already been well answered, and in particular several of the correct answers referred to what you call "charge buildup" in the comments where you say:

The "charge buildup" theorem seems to be a point of disagreement among the many answers, while not much information exists on this concept anywhere else.

The "charge buildup" answers are correct and there is a lot of information in the literature about this concept. It should be noted that the "charge buildup" is called "surface charges" in the literature.

Perhaps the seminal paper on the topic is Jackson's "Surface charges on circuit wires and resistors play three roles". This paper describes how surface charges act "(1) to maintain the potential around the circuit, (2) to provide the electric field in the space outside the conductors, and (3) to assure the confined flow of current". In particular, your question is most focused on (3) with some overlap with (1).

Essentially, as others have said, at very short times the currents and the fields are not well-described by circuit theory. During that time the fields act to redistribute charges such that there is a non-uniform surface charge density which acts to provide the local forces needed to "steer" the steady-state currents into the patterns described by circuit theory.

Although Jackson's paper is the most famous on the topic, my favorite paper is Mueller's "A semiquantitative treatment of surface charges in DC circuits". That paper provides a method for graphically approximating the surface charge density in a rough semi-quantitative fashion. The graphical procedure helps build intuition for where surface charges will accumulate.

The basic idea is that the equipotential lines are continuous, including at the surface of a conductor, but they can have sharp bends at that surface. The angle of that sharp bend is proportional to the surface charge density. By graphically drawing equipotential lines and looking at how they bend at the surface you can determine the regions where there will be the greatest surface charge density. Specific hints are given for drawing the equipotential lines.

One other important concept mentioned Mueller's paper is the fact that inside a circuit, where you have a meeting of two conductors of different materials, you can get a surface charge. In other words, surface charges can occur inside a circuit where you have contact between the surfaces of two materials.

This specific type of "internal" surface charge is particularly important for your question since it is this type that prevents the charges from flowing through the higher resistance in your question. At the boundary between the highly conductive wire and the resistor there are surface charges which oppose any current flow into the resistor and effectively steer the current around. This is how the current "knows" where to go.

This question has already been well answered, and in particular several of the correct answers referred to what you call "charge buildup" in the comments where you say:

The "charge buildup" theorem seems to be a point of disagreement among the many answers, while not much information exists on this concept anywhere else.

The "charge buildup" answers are correct and there is a lot of information in the literature about this concept. It should be noted that the "charge buildup" is called "surface charges" in the literature.

Perhaps the seminal paper on the topic is Jackson's "Surface charges on circuit wires and resistors play three roles". This paper describes how surface charges act "(1) to maintain the potential around the circuit, (2) to provide the electric field in the space outside the conductors, and (3) to assure the confined flow of current". In particular, your question is most focused on (3) with some overlap with (1).

Essentially, as others have said, at very short times the currents and the fields are not well-described by circuit theory. During that time the fields act to redistribute charges such that there is a non-uniform surface charge density which acts to provide the local forces needed to "steer" the steady-state currents into the patterns described by circuit theory.

Although Jackson's paper is the most famous on the topic, my favorite paper is Mueller's "A semiquantitative treatment of surface charges in DC circuits". That paper provides a method for graphically approximating the surface charge density in a rough semi-quantitative fashion. The graphical procedure helps build intuition for where surface charges will accumulate.

The basic idea is that the equipotential lines are continuous, including at the surface of a conductor, but they can have sharp bends at that surface. The angle of that sharp bend is proportional to the surface charge density. By graphically drawing equipotential lines and looking at how they bend at the surface you can determine the regions where there will be the greatest surface charge density. Specific hints are given for drawing the equipotential lines.

One other important concept mentioned Mueller's paper is the fact that inside a circuit, where you have a meeting of two conductors of different materials, you can get a surface charge. In other words, surface charges can occur inside a circuit where you have contact between the surfaces of two materials.

This specific type of "internal" surface charge is particularly important for your question since it is this type that prevents the charges from flowing through the higher resistance in your question. At the boundary between the highly conductive wire and the resistor there are surface charges which oppose any current flow into the resistor and effectively steer the current around. This is how the current "knows" where to go.

So, focusing on the resistor and specifically on the “interface” charges. Suppose initially that the current is too high (ie the current doesn’t “know” to avoid the resistor branch). This too-high current will lead to a depletion of positive charges from the entrance surface and an accumulation of positive charges at the exit surface. These surface charges will produce a field that opposes the current and reduces it. The charge will continue to accumulate until the current has been reduced to the steady state value.

This question has already been well answered, and in particular several of the correct answers referred to what you call "charge buildup" in the comments where you say:

The "charge buildup" theorem seems to be a point of disagreement among the many answers, while not much information exists on this concept anywhere else.

The "charge buildup" answers are correct and there is a lot of information in the literature about this concept. It should be noted that the "charge buildup" is called "surface charges" in the literature.

Perhaps the seminal paper on the topic is Jackson's "Surface charges on circuit wires and resistors play three roles". This paper describes how surface charges act "(1) to maintain the potential around the circuit, (2) to provide the electric field in the space outside the conductors, and (3) to assure the confined flow of current". In particular, your question is most focused on (3) with some overlap with (1).

Essentially, as others have said, at very short times the currents and the fields are not well-described by circuit theory. During that time the fields act to redistribute charges such that there is a non-uniform surface charge density which acts to provide the local forces needed to "steer" the steady-state currents into the patterns described by circuit theory.

Although Jackson's paper is the most famous on the topic, my favorite paper is Mueller's "A semiquantitative treatment of surface charges in DC circuits". That paper provides a method for graphically approximating the surface charge density in a rough semi-quantitative fashion. The graphical procedure helps build intuition for where surface charges will accumulate.

The basic idea is that the equipotential lines are continuous, including at the surface of a conductor, but they can have sharp bends at that surface. The angle of that sharp bend is proportional to the surface charge density. By graphically drawing equipotential lines and looking at how they bend at the surface you can determine the regions where there will be the greatest surface charge density. Specific hints are given for drawing the equipotential lines.

One other important concept mentioned Mueller's paper is the fact that inside a circuit, where you have a meeting of two conductors of different materials, you can get a surface charge. In other words, surface charges can occur inside a circuit where you have contact between the surfaces of two materials.

This specific type of "internal" surface charge is particularly important for your question since it is this type that prevents the charges from flowing through the higher resistance in your question. At the boundary between the highly conductive wire and the resistor there are surface charges which oppose any current flow into the resistor and effectively steer the current around. This is how the current "knows" where to go.

This question has already been well answered, and in particular several of the correct answers referred to what you call "charge buildup" in the comments where you say:

The "charge buildup" theorem seems to be a point of disagreement among the many answers, while not much information exists on this concept anywhere else.

The "charge buildup" answers are correct and there is a lot of information in the literature about this concept. It should be noted that the "charge buildup" is called "surface charges" in the literature.

Perhaps the seminal paper on the topic is Jackson's "Surface charges on circuit wires and resistors play three roles". This paper describes how surface charges act "(1) to maintain the potential around the circuit, (2) to provide the electric field in the space outside the conductors, and (3) to assure the confined flow of current". In particular, your question is most focused on (3) with some overlap with (1).

Essentially, as others have said, at very short times the currents and the fields are not well-described by circuit theory. During that time the fields act to redistribute charges such that there is a non-uniform surface charge density which acts to provide the local forces needed to "steer" the steady-state currents into the patterns described by circuit theory.

Although Jackson's paper is the most famous on the topic, my favorite paper is Mueller's "A semiquantitative treatment of surface charges in DC circuits". That paper provides a method for graphically approximating the surface charge density in a rough semi-quantitative fashion. The graphical procedure helps build intuition for where surface charges will accumulate.

The basic idea is that the equipotential lines are continuous, including at the surface of a conductor, but they can have sharp bends at that surface. The angle of that sharp bend is proportional to the surface charge density. By graphically drawing equipotential lines and looking at how they bend at the surface you can determine the regions where there will be the greatest surface charge density. Specific hints are given for drawing the equipotential lines.

One other important concept mentioned Mueller's paper is the fact that inside a circuit, where you have a meeting of two conductors of different materials, you can get a surface charge. In other words, surface charges can occur inside a circuit where you have contact between the surfaces of two materials.

This specific type of "internal" surface charge is particularly important for your question since it is this type that prevents the charges from flowing through the higher resistance in your question. At the boundary between the highly conductive wire and the resistor there are surface charges which oppose any current flow into the resistor and effectively steer the current around. This is how the current "knows" where to go.