I can barely follow you unfortunately. Maybe I'll start to get it as I quote and reply...

We can see that as r increases, U decreases. There is no limit to the bigness of r, so no limit to the smallness of U. What we're really saying is that if r is arbitrarily large, then U is arbitrarily slight, which is perfectly reasonable.
Yes this is very much reasonable. Basically 1/x... just don't divide by zero.

Therefore there is nowhere that the 2nd derivative is 0, and therefore there is nowhere that the curvature is 0.
I have no idea what you mean by this, but I like the conclusion. I'm ok with expansion into infinity, but I see the universe as finite. Like a Koch snowflake...



What's the perimeter of a Koch snowflake of a given area? Knowing it's finite, despite expanding into fractal infinity, is actually quite profound. That's how I see universal expansion... infinity bounded.

So. Your statement that "what is the top of the hill is a matter of perspective" is boneheaded UNLESS you're talking about observers moving at relativistic speeds, and in that case, it's a doozy.
Well why is moving at "relativistic speeds" important? I'm being technical and pedantic here. Motion at 1/1000000000th of c is motion, and subject to fractional time dilation. You can say it's negligible if you wish, but even the slightest bit of curvature is curvature.

Now, a reasonable definition of an equilibrium position is one where the slope of the field is perpendicular to the force caused by that slope. This is nice, as it means that we can definitively say that an object at rest w.r.t. the field is invariant to all observers, as predicted by GR. I.e. no observer should see the object moving w.r.t. the field, no matter how that observer is moving or accelerating.
I really don't like the term "at rest". Everything, literally everything, that has mass, is in motion. The idea that something is "at rest" might be useful when it comes to doing sums, but it's misleading imo. And it could be partly to blame for the contradiction you seem to face.

I have to be honest though, I'm still nonethewiser, I don't understand your contradiction.