e4 has more "forcing" lines compared to d4 imo
Comparing e4 to d4 - How much to remember

black is certainly just worse after d4 c5.
I mean, this is the guy that thinks 3. e4 outright refutes the Budapest gambit so I'm not sure he gets why

Personally I don't like playing dubious openings like the King's Gambit just for the sake of avoiding theory.
Right, I have only just realized that:
That's basically it - it's over. I saw how the computer handles it, by comp (-a heartless machine) it's all good, but you need to play really strange stuff - Kd1, Kc2...

E4 openings usually have way more theory on them than d4 openings. Sicilian and kings pawn opening is probably the 2 most studied openings that there are. However, if you wish to get as little opening theory as possible, probably English or 1.g3 is your best bet.

How deep you want to go, theoretically, depends on the lines you choose.
You could play both 1.e4 or 1.d4 in an extremely "narrow" fashion, if you wanted to, by avoiding mainlines and focusing on systematic sidelines.
With 1.e4, for example, you could transpose into the King's Indian Attack against nearly every defense black plays, using the same setup every game:
Really, the only departure you'd have to prepare for is 1...d5. Against everything else, you can just use the above setup. Not the most ambitious approach ... but still playable.
There are systematic approaches to 1.d4 also. I played this setup for quite a while:
You can play that against nearly any 1.d4 defense. Then you just figure out where you want your c-pawn to be, and your queenside pieces.
Very little theoretical learning required (though, again, it's not the most ambitious approach.)
But the point is: white can play the opening rather simple and easy, in both 1.e4 or 1.d4. You don't "have to" learn deep mainline theory, if you don't want to.
(Although, I do recommend learning some proper theory at some point, as it can all be quite instructive.)
If you play e4, you need to study way more, because there are more lines, and those lines go deeper, for example, think of the Open Sicilian versus something calmer in d4 territory, such as the QGD.