The short answer to why Dimorphos now has a 10,000 km long tail, some two…
Wikipedia claims that satellite orbits verify Einstein’s formulas for time dilation because the exact time and position of satellites are continuously adjusted. Their article makes it sound like Einstein’s equations are used to calibrate clocks and fine-tune positioning.
However, this cannot possibly be how things are done because there are other, more serious issues to consider, and these issues are of a kind that cannot be predetermined. Hence, adjustments to satellite orbits cannot be pre-calculated. They must be based on synchronization with Earth based stations.
Satellites are subjected to a wide range of factors that would make them go off into space or drop down to Earth if not adjusted for. Unlike orbits of planets and moons, there’s no self regulating mechanism that kicks in to keep satellites in place. They must regularly adjust their orbits, or they will be lost.
There is for instance a solar wind that pushes things around. Like any other wind, it’s strength varies in unpredictable ways.
Earth’s gravitational field is not smoothly distributed across our planet. There are gravity anomalies that makes Earth’s gravitational field lumpy and irregular. Earth’s gravitational field is not even constant. It varies in 5.9 year cycles.
There’s also the fact that Earth’s rotational speed is irregular. It speeds up and slows down in unpredictable ways.
Continents drift. Earth may even be expanding. There are irregularities for which we have no good explanation.
None of this can be pre-calculated. The only practical solution is therefore to use ad-hoc algorithms based on feedback from known positions on Earth.
It follows from this that satellites do not verify time dilation directly, because any issue related to this phenomenon is taken care of by ad-hoc synchronization routines rather than careful calculations. Any verification that there may be is indirect, based on data collected from satellite logs.