Ive never had much of a problem with the tailstock moving while tightening collets. Occasionally it might move a bit, but its never resulted in a slipped spanner yet. Crap under the tailstock happens a lot on hercus's anyway without way wipers fitted to the tailstock. Thats actually something I want to make one day, along with something to retain the tailstock down a little better than the cam / nut arrangement.
Holding the job with a rag while parting isnt ideal, but youre gunna have to suggest a better way to catch a polished sphere if you want to level that as a criticism. Also, it was paper not fabric, which should be a bit safer (but still not ideal).
Yes, stopping the chuck with my hand is unsafe. The likelihood of stopping that in my home shop is slim. I have considered fitting a spindle brake to the back of the chuck as a project one day. ill try not to do it in videos, thats a valid point.
The use of the countersink tool to remove the horrid burr off the back of the part wasnt great either. What I should have done was regrind the parting tool to cut on an angle to help eliminate it. Part of it too is that I was parting into a thread, so the parting tool removed enough material until the thread was weakened enough to break apart. I should have left material untapped on the mandrel section, made a pointy parting tool. Alternatively, I should have made up a 3/8 mandrel and avoid the whole two threadforms altogether. I could have put the ball back on a mandrel and used the lathe to deburr it properly, but I was under a good deal of time pressure to get content out there.
The compound didnt move when I bumped the shoulder, the toolpost did. The school lathes use lantern style posts. If Id cleaned under it like you suggested, it might have helped a bit. Tightening it properly would have made all the difference, it really was barely done up. Its easy to get distracted from proper setup when youre under time pressure, trying to film with a child at home still and filming for the first time.