To be clearer, if an object falls down a well hole in a massive object,
starting from an infinite distance above that well, then its velocity down the well will be the escape velocity from any position in that well. But, if it starts falling from zero velocity on the surface of that mass, then its velocity in the well is much below its escape velocity from the mass - if there is no frictional losses, then the velocity of the object starting at rest on the surface would only be enough to get it back to the surface if the velocity could be reversed.
And, it is true that the time dilation in the vicinity of a mass is directly related to the escape velocity (to infinity) from that mass. Which is one reason it seems almost like space is flowing into the mass, since the calculation for the time dilation is the same as looking at the differences in the rate of time passage at two different velocities associated with 2 different distances from the mass with 2 different escape velocities. But, mentioning that concept really sets off the theorists.
Anyway, yes, Questioner, what I would like some experimental verification for is the assertion that time dilation follows escape velocity, rather than local gravitational acceleration, at points
inside a massive object. They all follow the same relationships outside of the mass, but that does not logically require them to follow the same relationships inside the mass. And, we are having trouble making our observations of velocities within the masses of galaxies fit our theories. So, why not do the experiment?
There is a school of thought that gravity doesn't "bend space" to make light curve, but rather it curves time to make light curve. See
https://www.askamathematician.com/2...ved-space-time-cause-gravity-a-better-answer/ for example.
How we tend to think about things is based on our experiences, and we are finding that our experiences are not a good intuitive basis for some of our theoretical predictions. I think we are really bad at understanding time, for instance.