The answer to your question is stress-energy. Both gravity and inertia are rooted in stress-energy.
In the case of gravity, it is more obvious, because we have general relativity, and one of its building blocks is that stress-energy is the cause of gravity, and everything and anything we know of that does possess stress-energy, does bend spacetime.
yes they do, and for the reasons you sketched out.
Do photons have inertia?
In the case of inertia the source being stress-energy is not so obvious. There is a nice example of this in the fact that even massless photons do have inertia. This is because they do have stress-energy, and so they do have inertia.
Symmetry of the Lagrangian with respect to translation in time and space (in classical mechanics), leading to conservation of energy and momentum.
The claim that the worldline of an object in free fall is a timelike geodesic of spacetime. (Such a worldline can also be described as a line of maximal proper time between any given pair of events on the line.)
Is there still no known origin of the law of inertia?
There are different ways to describe the origin of inertia, but most on this site agree that it can be deduced from conservation of energy and momentum, and geodesic motion.
The answer to your question is that since photons do have stress-energy, and this causes them to both bend spacetime and have inertia at the same time, proves (or gives us a hint) that both gravity and inertia are rooted (even if in the case of inertia the connection is non-trivial) in stress-energy. This could lead us to the expectation that gravitational mass and inertial mass should be equivalent. And then this is experimentally (based on the equivalence principle) proven.