Sounds like a reasonable variant.
But I'm confused why you say that 2-player in particular didn't impress you: it sounds like you don't like that you only need to look at your own stack and the draw stack... But that's true in multiplayer as well as in 2-player! Why does it (apparently) not bother you in multiplayer that you only need to look at your own stack and the draw stack?
Actually I haven't had a chance to play multiplayer yet! But honestly, I don't think I will particularly like any of the multiplayer versions except for Hot Potato and maybe the Triplets. edit: I forgot about Poisoned well!
Even with the Triplets, I fear it seems to be just a weak version of Set...and I'm actually thinking of making it "Quadruplets" for that reason. Or maybe add a restriction where the three cards must be in a line "Tic-Tac-Toe Triplets".