I like to do ~10 minutes of easy puzzles before the first round of a tournament for the same reasons you give. Just "warming up" so to speak. I think a few blitz games would work just as well (probably not G/3 though, too fast I think).
But my openings are, I'd say, 99% of the times less prepared than my opponents in a tournament and from time to time I've experienced similar results. I also think it's because I've been fighting though variations and such while they've been playing memorized moves so when we reach a middlegame I'm "warmed up" to the position. That or they think I'm no good and don't try as hard until the position evens up.
I have over a number of games noticed a trend.
I get inferior positions out of the opening. Not losing positons by any means. But positions where after about 10 moves an engine gives me as half a pawn worse or so.
Then in the middlegame I start playing more accurately, find some resource that gets me back into the game etc, and get the middlegame back to equal, then better for me, then winning, and then my opponent resigns.
This has happened time and time agains at long time limits against good [1900 and better] players.
It could be a lack of opening theory. Myabe. So I have worked on that. But I also wonder about aspects of chess psychology. Maybe I take about 10 to 15 moves to "get my eye in" or "Warm up". Maybe it's the adrenalin of starting a game and I move a touch too quickly at the start. I don't know.
I have wondered more than once if I should play a warm up game of blitz about 15 mins before. That would be ironic as I am not a blitz person. Or maybe even a chess puzzle 10 minutes before.
I am like a boxer who has pretty average first and second rounds but who "warms into" the contest and by round 5 or so I am positively dangerous
Are you dealing with your chess psychology/mentality in relation to how it goes? [and if your name is the Michael Douglas guy from Basic Instinct please spare me a long lecture on the stupidity of shrinks :)]