Steps to take after buying an Omega on eBay?

I'm going to say it's a combination of gravity and momentum in practice. On the wrist, there are some rapid direction changes and also some re-orientations with respect to gravity, I'm guessing you would still get some winding in microgravity, but it would be less efficient. As an aside, most winders rotate at a slow constant angular velocity and the rotor motion is mostly due to gravity, IME. If you oriented a watch winder so that the rotor axis was vertical, it wouldn't be very effective.

IIRC, NASA deliberately specified hand-winding, but not necessarily because of the micro-gravity.

As a timing check for a manual watch, it does make some sense to put it on a winder since it changes the position. However, it's a relatively small sub-set of full positional variations, TBH.

In addition to wearing the watch and testing the chronograph, I recommend registering for a free account on Omega, adding the watch to "My Watches", and getting the test results specific to your watch based on your serial number.

I think Omega will honor the warranty regardless of where you bought it as long as you have the warranty card so I don't think there's anything to worry about there.