Does this make sense?

That some people's "best games" are just lost games they happened to win on time?

I think if a player wins on time, they should have the option of taking points or not. I have won a few on time that could have gone either way. Makes sense to me.

 Me too. But a lot of them are games I had won, but they were just too crap headed to resign.

My "best" game was finished in the position below:

I really want to beat a higher rated oponent just to change this.

The rating system has to be the same for everyone, otherwise you would take your unreliable 1400 rating up against 1400 strength players and soundly beat them. The system to find competitive games would be destroyed.

What works better is for you to be able to drop your time out wins from "best game".

I see II Oliveira's point, his "best game" (which should be called "highest rated opponent beaten by check mate") was really not deserved. If anything, most people would just deny points won from time and therefore the game just didn't happen. I also see woodshover's point that some players know they are losing and don't want to go through the "agony of defeat". Well, then just resign. Come on, it's just a game.

 Then perhaps that's what it should say.

Or perhaps you could have the option of rejecting it as your best game.

 I don't think that rule should be disregarded, but some of those best wins are kind of insulting. Especially in turned-based. Sometimes the poor slob has nothing but a rook, and a few pawns, with their opponent having most of their material, but for some reason just didn't bother to finish the win. Though of course it should be on the record as a win, it looks kind of silly being called the best win.