1. e3

Of course you can play it - there are many other systems one can transpose into. But why wouldn't you just push one of the centre pawns 2 forward instead - just a better version lol

If I want to stay flexible I prefer the Reti. Nf3, c4, double-fianchetto and only then moving center pawn to e3 for example.

I play this Nf3-b3-Bb2-c4-e3 stuff from the game linked in #2 a lot, but I always start with 1. Nf3.
Can't see any benefits of the 1. e3 move order, it just seems to give black more options.
Given the huge rating difference, that was probably the point, though. Just let black lose some more minutes in the opening...

@Dr_King_Schultz 1. e3 is a lot more flexible for white (for example white can transpose to the French Exchange or the Queens Indian Defense or the Reti Opening). 1. Nf3 is equally good.

It's more flexible for black.

White cannot transpose to any of these openings - it's black who does!

Alkehine played this opening only 1 time and win ( at simul of course)

@Dr_King_Schultz If you play 1. e3, you transpose to the french defense exchange variation, queens indian defense, reti opening, or queens gambit lines. If you play 1. Nf3, you can transpose to the French defense two knights variation or queens indian defense or queens gambit lines. So, white gets equal flexibility with 1. e3 and 1. Nf3. And also, 1. e3 has the benefit of being a rather unexpected opening for most people.

„Unexpected“ and „interesting“ means most of the time: bad and you cannot play it twice against the same opponent. ;)

1. Nf3 e6 2. e4 d5 3. exd5 exd5 and there's your beloved French Exchange. Nothing "lost" by the new move order.

The question is not which openings could "happen", but who can decide / force it.

And in that regard, 1. Nf3 is simply better - unless you desperately want to play a reversed Dutch against certain setups. (Or if you want to troll your opponent, which may be a valid strategy if your outrating him by several hundreds of points, as in the Carlsen game someone linked here.)

1. Nf3 takes away 1...e5 as a choice for Black (well, at least makes it a lot less sound).
2. e3 takes away nothing from Black.

On the other hand, 1. e3 takes away more choices for White than 1. Nf3 since in general there are a lot more lines with Nf3 + without e3 than with e3 + without Nf3 (and in many cases where White does not play Nf3, it's because he wants to play f3 + e4 instead, which doesn't quite work with 1.e3 either).