The chain on my 1yr-old M6R broke yesterday so I decided to put on a new chain (same size and spec)

However, when I put the old chain next to the new one to count the links, I noticed the old one was longer - About 0.5cm over the length of the chain. I understand this is because the chain is worn (which is normal for 2-3k)

Anyway, I put the new chain on, and it started skipping - Could this indicate wear on the chainring or sprockets? - This has me confused as first I thought I'd get 10k out of the chainring/sprockets and also, they don't look worn to me - I'm looking for the 'shark-fin' pattern on the teeth, but can't see it.

In the end I fixed the old chain and put it back on.

Does anyone have any experience of chainring/sprocket wear on a brommie, and what it looks like?