>>1894176hard problem to diagnose, I apologise in advance for the TLDR text wall
does it skip horizontally (axial) or vertically ? (radial)
Horizontal is normally to do with cable tension / friction and binding, in some rare cases the bike frame twisting under load
it also ties into derailleur non-alignment and to what extent the pivots are worn loose,
loose pivots can also affect limit screw effectiveness, .eg the part of the Derailleur that hits the limit screw is not the part that is guiding the chain,
hit a bump and the Guide plates can wobble or bend further than if they were new parts.
vertical in your case might be that the chain does not have enough tension, on extremely worn Rear derailleur the P knuckle spring gets too weak
its rare for the B pivot Spring to get weak, but it's bearing/bushing does wear out sometimes, resulting in wobble
I dont get the impression that the return springs on yours are weak as its very uncommon, but its still worth checking
only other thing that comes to mind is chain wrap, your new chain might have too many links ?
I would be surprised if a new chain and Cassette were incompatible, the chain pitch and width should be suitable
and cassette spacing is a lot more standard than it used to be
https://www.bikeforums.net/early-brifter-bikes-7-8-9-speed/1234877-8-speed-cassette-spacing.html>11t and 13twew lad, I'd check the B screw, as its somewhat important in following the profile of wide range cassettes
https://wikipedalia.com/images/3/3d/Sprocket-slope-sm.jpg 
