In thermo-statics or thermo-dynamics, you should think of the "surroundings" as being the boundary conditions for the equations. Their role is very similar to a hypothetical ideal battery or voltage/current source in an electric circuit with resistors, capacitors and inductors described by Kirchhoff's laws. In real life, there is no such a thing as a source whose voltage is independent of its load and is also independent of temperature, pressure, humidity, etc., in which it is imbedded, as anybody trying to start an engine in the winter learns it. Of course, if we did not make such idealization we would not be able to calculate the current through even the simplest linear resistor...
An ideal battery has zero internal resistance, has infinite amount of freely moving electric charge, and can absorb from or release in the circuit to which it is connected any amount of said charge all at the same voltage.
An ideal thermal reservoir has infinite amount of internal charge called entropy, its temperature is independent of anything the reservoir is attached to and with it can exchange any amount of thermal charge, i.e., entropy, always at the same temperature. Notice the analogy with an ideal battery. From the point of view of the thermal reservoir entropy is like electric charge and temperature is like voltage. In the case of a battery the simplest model to the load-dependent voltage drop is a non-zero internal resistance that may depend on its environment (temperature, pressure, humidity, age, etc.). For an ideal thermal reservoir connected to a thermodynamic system one may employ a similar "internal resistance" by specifying the interface between it and the system to be a thermal conductor whose one end is at a fixed temperature as defined by the ideal reservoir but its other end is connected to the system and its temperature is variable.
The analogy is not superficial, it only fails in the sense that electric charge is unconditionally conserved always and everywhere while entropy is only conditionally conserved. Total entropy along with entropy change in detail is conserved in a reversible process but is only "semi-conserved", that is lower bounded in an irreversible process.
As you can imagine such idealizations can lead to apparent paradoxes, contradictions, etc., unless one is careful how to use them. For example, one may not ever directly connect two ideal batteries of different voltages in parallel. The battery itself has infinite energy and infinite charge, etc., so one cannot apply energy conservation to the complete circuit including the battery and thereby calculate the total energy, only energy changes are meaningful when an ideal battery is included. Exactly the same way one may never connect two reservoirs of different temperatures directly, only energy changes are meaningful not total energy, etc.
And finally, one may go beyond the concept of thermal reservoirs and introduce work reservoirs. These are similar boundary conditions to temperature but for the mechanical, electrical, magnetic, etc., intensive parameters, such as pressure, tension, stress, gravitational potential, electric potential, magnetic potential, etc. It is assumed that the reservoir holds an infinite amount of the corresponding extensive quantity, that is charge: volume, density, concentration, mass, etc., all with the underlying assumption that the finite amount of charge that is exchanged with the system does not effect the potential itself at which the exchange takes place.
Having said all this it should be clear now that the process in an ideal reservoir must be reversible irrespective of what happens to the system it is connected because this is how we set it up. In the mechanical reservoirs there is nothing "thermal", it has no temperature nor entropy, everything is reversible by definition. In an ideal thermal reservoir we cannot define the increase of the total entropy for that is infinite but we define the reservoir in such a way that whatever entropy is absorbed at its boundary is what goes in it, and whatever leaves its boundary is what goes in the system to which it is connected, all this by definition, so there again the entropy generation if there is one happens outside the reservoir, again by definition, so the reservoir's processes are reversible.