Why does thermodynamic integration work?

Some mathematical statements then an intuition statement:

Mathematical

We often use parameterizations like this to mathematically represent a continuous motion from one point to the next, for instance when discussing convex spaces, in which we say a space is convex if all points between any two given points are also in the set (in the language of the wiki page, convex if the vector $\lambda u_i+(1-\lambda) u_j$ is also in the space, for any vectors $u_i$ and $u_j$ in the space; satisfied by a sphere, but not by a donut).

Similarly, the parameterization here is simply a way to represent traveling through a space, in this case a space of possible states. The "nonphysical" parameter $\lambda$ isn't introducing new physics any more than my above analogy means the space of a sphere is in any way physically changed by our mathematical wandering through the space of a sphere. The reason this works is that you're wandering through a continuously changing energy landscape, which brings me to...

Intuition

For state variables like energy, as long as you wander from one state to another, keeping track of your progress as you go (more on that below), you can find your new energy from your old, just like with the traveling analogy already discussed.

The nuance is that if you, say, travel from one city to another city directly North, you might take a path that veers a bit east before coming back west to arrive at its destination. You might say, doesn't that increase my total distance traveled? Yes, but any travel east cancels any travel west. Similarly, going through this space of states with varying energies, any increases along the path cancel any decreases along the path, and you can arrive at the total energy difference between your initial and final states. Now you can see why we need this path to be differentiable with respect to $\lambda$: we can't allow discontinuous jumps.

So in conclusion, we don't really care about the nature of the path, since any funny deviations from point A to point B will cancel. $\lambda$ is just the mathematical parameter that allows us to continuously follow our path and ensure we end where we want to go.

I will preface this by saying that I have not heard of what you are describing in such a generic context. But I do not think it applies to every thermodynamic variable, rather it applies to only those that are path independent such as state variables. It would not apply to path dependent variables such as heat and work (which may be path independent if the process is reversible).

You have many constraints on what parameterizations you may use. It must satisfy the boundary conditions -- this is very important. State variables describe what is happening at equilibrium. Equilibrium means (broadly speaking) that there are not more changes occurring to the system. So the only values that are important are the end-points. This is why your parameterized function must match at the end points of the integration. And because it is only important at the end points, it doesn't matter how you get there. You'll get to the same equilibrium.

Maybe a real world analogy would help. I live in Atlanta and I want to get to New York City. My current state is Atlanta -- I am at equilibrium here. My future state will be New York City. I will be at equilibrium there.

I can choose to drive on the interstates. That's one path. I can also take back roads the whole way, that's another path. I can fly. Maybe I fly direct, or maybe I need to connect in Charlotte, or Dallas, or Chicago. Those are all different paths. So I can parameterize my transport many different ways. But because all of my parameterizations start in Atlanta and end in NYC, they are all valid integrals to move my state function (location) from one endpoint to the other.

And it should be obvious why work or heat is path dependent and cannot be parameterized this way. The amount of time and energy consumed in fuel is vastly different depending on my path. So those are not state functions and the parametization is critical to getting the correct measurements of those.

But my start and end states? Not important how I moved between them, just that all my paths begin and end where they need to. And we know the difference in miles between Atlanta and NYC, so we know the difference between the two equilibrium points. How far apart they really are is independent of how I decided to travel between them.