Shouldn't there be a race on both sides of the rear wheel bearing?

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Do you mean spacer?

I'm sure mine didn't have spacer(s), the the bearing already has a race built in that it runs on.
There is a washer, but that goes under the big nut.

The first pic looks like it has a ding/burr, file that down flush.

The outer race is part of sealed bearings.

But Eric has a video showing how to remove the race

Often when you knock the old bearing/hub out the bearing busts apart and the old race is left stuck on the hub.
Your new bearing has a race already.

The race usually separates from the sealed bearing upon diassembly

So it's part of this?

Click to view attachment

Yep