Wanted to circle back and say thanks to those that responded.  This thread has indeed got me interested to prepare against the Center Game too (and I'm going to go with the ...Be7 plan instead of ...Bb4, regardless of Bd2)

MartinC wrote on 04/26/23 at 08:03:30:

Comparing that to some ancient theory, 1. e4 e5 2. d4 ed 3. Qxd4 Nc6 4. Qe3 Nf6 5. Nc3 Be7 6. Bd2 d5 7. ed Nxd5 8. Nxd5 Qxd5 9. Ne2 0-0 was given as equal by Nunn in NCO.

3.h3 is not terrible. I've done this sort of thing myself, the idea is to get both players thinking as early as possible. On the other hand, it also gives black a free choice. I would be inclined to choose 3...e5 with a reversed Center Game. That should be equal already, and black can try to find a variation where the extra h2-h3 is a weakness.

George Jempty wrote on 04/24/23 at 22:04:05:

1. Surprising the surpriser is debatable. It can work out if you have two lines prepared, the first for regular use and the second for variety. When an opponent switches to an unusual (for them) opening to catch you in preparation for the first, then it's a good time to wheel out the second. But if you consistently reject principled moves because "they would be expecting that", that's a recipe for getting a lot of dodgy positions.
2. I agree wholeheartedly, the only problem is in knowing where the "necessary" line is. You may think you have found it, but then your opponents will occasionally show you that you erred a little too far towards the "least".

I've been playing 3.Nf3 the last half-decade or so but 3...Bg4 probably equalizes but now I've hit upon the idea of 3.h3!?  After 3.Nc3 (obviously best but I don't want to have to learn 3...Qa5, 3...Qd6 and 3...Qd8 though against the latter why not 4.h3!?) and besides 3.Nf3 and 3.d4, SF-15 NNUE, so better with regard to openings than ever before, ranks 3.h3 as in the top 3 or 4 choices. 

I've only found a couple of master games, Black responded in both with the not so-challenging 3...Nf6 and after 4.Nf3 White may retain a small opening edge.  And after 3...c5 I plan to respond with 4.Na3!? possibly reaching a formation I'm quite comfortable with playing the 2.c3 Sicilian

tldr; (below: ignore)

I know that 3.h3!? isn't objectively optimal but my repertoire has two guiding principles: 1. Surprising even the surpriser and 2. the least necessary preparation -- I think memorizing games from Tal-Botvinnik 1960 and playing (the beatable) Fritz 2.51, or a 1990's console *blindfold*, does more to advance my chess ability

Sorry for the  above, it's just that I've sometimes gotten responses like "learn the main line", so I wanted to intercept such