I don't see any insects or creatures such as you describe.

"I'm entirely new to growing clematis, or much anything for that matter. I got the clematis a couple of months ago, and it's been growing indoors in a self-watering planter."

This is probably your primary problem........clematis are NOT houseplants and will not grow well or happily indoors. That alone can account for the discoloration of both leaves and flowers.

Thanks for the comment! They are admittedly tiny, I wish I had a proper camera. I've marked some of them here, does that help?

There does seem to be a clear connection where the leaves that have this damage to them are also the ones that have these creatures running around on them.

Can’t see detail. Might be thrips. I’d take it outside and hose it off.

tj

Thanks for the tip. Reading about thrips they indeed could match what I'm seeing, although they are too small to tell for sure.

Don't have a hose, but I gave it a forceful spraying with some garlic water (apparently kills bugs of all kinds), and will leave it on the balcony for a while now.

If I manage to eradicate the bugs, could the damaged leaves recover, or should I just prune them all?

The leaves are not damaged enough to warrant removal. They are still doing their job of photosynthesis. And I would suggest you leave the plant on your balcony. It won't last long indoors.