50 years ago we would replace the cogs and chain at the same time. The chain does not stretch, but wear on the rivets and bushes makes it longer, so runs higher up the teeth and wears them. A new chain then tries to run where the previous longer chain ran, and then slips. This would only happen on the small cogs. The wear on the chainrings is unlikely to matter. I wonder if the chain is too wide, and catching on adjacent teeth. In that case it would run only on the largest cog.
Keith
As everyone else has pointed out your rear block is worn out. Not so much on the smallest cog, but if you compare the teeth on the small cog to the others you can see how worn they are.
The derailleur should have a retaining bolt and small plate which fits into the drop out and stops the derailleur twisting as you shift gear. I'd dump the derailleur and buy one of these:
Drill out the original gear mount and refit a new derailleur
EDIT* Just read the first part of your post and realised that's what you are planning :facepalm:
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