I haven't used the 2DS, myself. I can't comment on any and all games for which 3D is "required."
The only one that I know of offhand is Super Mario 3D Land. And the 3D requirement is only a few occasions here and there.
There are a few spots where you go down a pipe into a little bonus area and you see test in the corner that says "3D". In one example of this, there is what appears to be a staircase of blocks behind you to climb. There stair case is in fact that, but when you switch to 3D mode, you see that what appears to be a block on the top of the stair case is in fact floating in the foreground and you have to jump from the stair/block pile to that block.
I suppose you could probably pull it off without the 3D, but I can see where it would be hard to figure out what to do if you aren't already familiar with it.
But otherwise the game can be played in 2D, and for most other games that I've played (which I admit is only a handful), the 3D is optional.
If you just want the cheapest way possible to play these games, and you don't care that the 2DS can't fold up like the previous DS/3DS models, it's not a bad way to go.
That being said, I just recently upgraded from the regular 3DS to a 3DS XL, and I'm LOVING the bigger screens!
It just makes things easier to see. For instance, I downloaded Super Mario Bros. 2 for NES a while back from Nintendo's virtual console store. I found it difficult to play on the regular 3DS screen, but on the 3DS XL, it's much better.
My
much younger sister will be inheriting my regular 3DS as she's been wanting one, and is fine with the one that I already had.
So, all of that said, I personally do recommend the 3DS XL for the bigger screens. I believe there will be some black friday deals coming up where you can get it for $150. But the 2DS still isn't a bad option. I think the 2DS screens are roughly the same size as the standard 3DS.