Reuse nail holes when replacing oak trim
Just two quick things.

Given that your nail heads were covered, I would have just bent them back and broke them off.   But if they came out and you have holes on the other side that's fine.   Honestly every trim/nail-type might have a best method so there isn't anything that someone can categorically say.

To go into existing holes... Get a better quality trim nail and use a gun.   Up the pressure on the gun.   When nailing into existing hole you want to hit the hole at an angle  - maybe 30 degrees - so that your head doesn't shoot straight through.