A minor piece odds is a win for sure, any chess play can tell you that. But a pawn up .. that is unclear. You speak of GM games being decided by a pawn which of course happens but what you do not speak of is material sacrifices in GM games which are common too. And there's all those endgames which the pawn-down player wins (often by superior play, sometimes other reasons). And there's all those well known ways an extra pawn in an end game can be a draw, including for example king+pawn vs king is drawn unless it's set up so you can force promotion quickly (e.g. defending king is out of position). King+rook+pawn vs king+rook is common and this is also drawn in general (the defense is called the philidor position) unless the pawn-up side starts in an advantageous position (something equivalent to the lucena position)
One of the interesting facts about pawn odds is that you now have an open file b/c of your missing pawn. And pawns restrict moving out your pieces so having one missing can save time. The worst thing you could do is take away black's f7 pawn at the start. That is, first guess, a loss for black. Take away some other pawn and do it from white and it's a lot harder to guess.
The rule of thumb taught to beginners is that a pawn is worth 3 moves in the early game (and a knight is 3 pawns so you might think it's worth 9 moves, but that conversion doesn't work well, with 9 free moves you could set up checkmate). I think a pawn is worth somewhat less than 3 moves but it really varies and the comparison doesn't entirely make sense.
BTW there do exist well known openings leading to an unclear/unknown result that involve a piece sacrifice (lines in the King's Gambit or Najdorf for example). Losing a piece for nothing is a clear loss but various kinds of compensation are possible.
There are so many exceptions and pitfalls to work out that I'm sure that there is no 'shortcut' to the answer, but I do lean to support the idea that a queen advantage should be a win, in fact, I think that any piece or even just a pawn advantage should be a win, witness how many grand master games literally turned to watershed losses once that precarious balance was lost by as much as a single pawn.
Even so, the risk a of a draw is significant, the risk of an opponents win much less so.
Chess is all about the 'mate', not about who has the most points on the board anyway. Woe the player that forgets this even for a moment.
It's funny how the '3 repeating moves' rule makes chess actually much harder to reason about in terms of guaranteed endings because it means that a definite material advantage may still result in a draw, is there a factor known for how much this rule adds to the complexity of proving a win?
Another question that might be interesting is if a pawn advantage would be enough to cancel whites advantage by starting the game, I suspect that a single pawn is worth more than whites advantage but I'm not sure about this, and it might depend on the pawn (some pawns gone would allow white to deploy much faster with the price of the pawn only appearing later on in the game).