It seems this issue has been fixed in snapshot 24w36a as part of the rewrite to equipment code. I can not recreate the issue in 24w36a and I looked at the code identified to be responsible in MC-274309 and it appears to be fixed.
Those are indeed the same issue, is there a way to mark as duplicate but add the reproduction steps and code analysis I provided? MC-248973 is the more inclusive report (it relates to all types of actions with this issue), so it's probably best to mark this and MC-159163 duplicates of that and move over the extra information.
Can confirm this is still an issue in 1.19.2. The provided reproduction steps are very situational but this bug seems to cause various other issues and be reproducable in other ways.
Specifically, I ran into this while trying to debug an issue where the second skin layer isn't shown when many players join a server at the same time. It runs into the same race condition which prevents the skin settings from being synced with some other clients. It's hard to reproduce but happens quite commonly with large amounts of players.
Thank you for looking more into this. That this change lead to the related issues was an assumption I made that I did not confirm and I should have phrased it as such. I've now modified the issue to reflect this.
Here is a version of the example resource pack that is properly zipped.