Not sure why being under oath is such a big deal? It's still pretty easy to lie and play dumb.
It's a big deal because sure he can lie under oat, but if he gets caught? That's the big deal. That's super illegal. And like, look at how this already played out just naturally:
- Did you have this meeting?
DTJ: No I did not
-Ok but here's proof of the meeting
DTJ: Ok I had the meeting but it wasn't about anything bad
-Ok but here's proof that the meeting was about something bad
DTJ: Oh well I uh *releases all the emails himself*
So imagine that happening but he's under oath. If he had been under oath and said there was no meeting, and then they countered with "well here's proof there was" and he changed his story and they countered AGAIN and he lied AGAIN he'd conceivably leave in handcuffs (or however it works in reality when everyone does their job properly and puts away people who lie under oath)