Do you think it would be good to start over now?

No, not really.

Even if they are outdated, I still want to kinda understand everything

Which you won't because they're outdated and you will level through the content within a week. You can't play through legacy content that lasted years in a way that gives you anything close to the same experience. Your artifact weapon is dead, the two years people spent leveling and upgrading it you will never be able to do.

Nothing else you listed require making a new character, neither would making a new character make those things any easier.

class halls

You can still do this with your main, you can go back and do your class campaign and quests, unlock your class mount while you're at it.

professions, pet battles

Both of those are independent of new characters, the game doesn't force you or push you into pet battles or professions while leveling. Both of these can be done on your main.

I maybe wait till Shadowlands release and the restructuring (aka streamlining) of expansions into Levels 10-50?

Right now you will outlevel an expansion before seeing a quarter of the content. So you can either do things you've outleveld on a new character or your current 120, but either way you're not going to experience all the content of an expansion the way you want too while leveling a fresh toon.