Er sieht ja so gut aus!

What’s the purpose of the “Ja” here? Would the sentence work without it?

Yes, it would work but it might change the meaning to a degree.
The ja does not only emphasise just how good he looks, its also a potential indicator of an ironic/joking undertone - though, without further context there’s no way to tell if such an undertone is present here as well.
ja as used in this sentence is strictly in spoken language.