Let us begin with a friendly reminder that, as per Einstein's Equivalence Principle, UA is should be locally indistinguishable from the RET gravitational model. This is, in effect, part of your own model (or at least an undeniable consequence thereof). By proposing that you could locally distinguish between UA and RET from just slightly above the Earth's surface, you are doing more to attack your own model than ours.
But let's consider the mechanics of a slinky, and figure out
why it's the top that falls first. I'll explain the issue in layman's terms, but if you'd like more detail, there are plenty of resources available online. I will also ask for some patience from our native English speaker lounge - I'm certain I got some of my terminology wrong. Mechanics is not my strongest field, and translating this stuff on the fly is tough business.
Though it ultimately won't matter, let's assume that the Earth's gravity operates exactly as it would in the Round Earth model for now. The answer is tension. As the slinky's springiness causes it to contract back to its normal state, we can easily illustrate the forces acting on both ends of the slinky like so:

These is just an illustration of how tension works on the slinky itself, with no account of any other forces. I'll glance over some details for the sake of brevity and assert that the scalar F
S1 is equal to F
S2 - a slinky is generally close enough to uniform, so naturally the tension would be distributed evenly.
Now, we know that the slinky was released by someone holding it off the top of a bridge, and that the slinky was at rest just prior to that. What caused it to expand? Right now, I'm assuming RET, so the answer is simple: weight. The slinky got pulled apart until the spring's internal forces were able to balance out its weight. If this were not true, we'd be observing the slinky either expanding or contracting prior to being released. In other words, we now know that at the time of the slinky's release F
S2 was equal to W. Again, since the slinky is largely uniform, weight will affect both ends of the slinky equally, like so:

So, why does the slinky fall top-first in a Round Earth model? We know that F
S2=W, and thus the bottom of the spring is in equilibrium at that very moment. Meanwhile, the top of the spring is affected by two downward forces. The top starts falling, but the bottom doesn't. The balance of forces changes over time, but the entire scene doesn't take long, so for most practical purposes the bottom doesn't move until the slinky mostly contracts on itself.
Right, so that's RET covered. How does FET change? Well, weight is (mostly) gone, so we're back to our first diagram - except this time the entire world is accelerating upwards:

We have to ask ourselves the same question we did before - how did the spring get to its expanded-yet-resting state? This time, the bottom of the spring was not being held by anything, and the Earth was accelerating upwards, together with the person holding the spring, and the spring's top. But not the bottom. Eventually, we reached a state where the spring's tension would cause the bottom of the spring to accelerate in tandem with the rest of the system. At this point it's more intuitive for me to think of the system in terms of acceleration rather than force, but the conversions are trivial, so please just bear with me.
We now know the following about our scenario:
- The bridge, person on the bridge, and the top of the spring are accelerating upwards at ~9.81ms^-2 due to UA
- The bottom of the slinky is also accelerating upwards at ~9.81ms^-2 due to the spring's restoring force
Immediately prior to the spring's release, we'd be looking at something along the lines of:

Where FX=FS1*2
- The spring is now released, so the top is no longer subject to UA (and, indeed, starts accelerating downward due to the same restoring forces), while the bottom remains at equilibrium
As expected, these situations are precisely analogous - it's just representing the same scenario from an inertial vs non-inertial frame of reference.
The error in the video is assuming that the bottom of the slinky is not accelerating together with the Earth - but we know it must be, it's a fundamental consequence of how the experiment was set up. After all, if these were not analogous, RET would be an impossibility and we wouldn't need to have this discussion
