I'm no expert on Chess, but I think about it this way:
It seems to me that in the opening position of FIDE Chess, most pieces are in quite poor positions, with their mobility greatly restricted by overcrowding and no immediate opportunities to threaten enemy pieces. An extra move can be used to develop your pieces into more advantageous positions, which translates into a higher probability of winning. As a corollary, the value of an extra move changes throughout the game based on how rapidly your position can currently be improved.
In On Numbers and Games, John Conway develops a theory of combinatorial games (Chess doesn't quite fit in this set, but it's close) as being a superset of numbers. Certain games (or sub-positions within larger games) have an exact numerical value because their existence gives a direct advantage to one player (positive) or the other (negative). But other positions have a "fuzzy" value; it's not clear how much advantage they give because it depends on how many moves each player takes (and in what order) trying to improve that particular sub-position rather than some other sub-position elsewhere in the game. He has a concept of "heat" that corresponds to the volatility of a position; the "hotter" a game is, the more advantage can be gained by whoever makes the next move.
(Note: I don't quite follow all of the math in this book and it's possible my description of his theory isn't entirely accurate.)
As for Betza's "quantum of advantage", I think his position was a little more complicated than that. He said that a third of a pawn was roughly the smallest advantage that a master player would notice in practice, and therefore that it makes a good approximation for lots of different minor advantages that are NOT actually equal but all roughly on that scale, and lists a tempo as one of several examples. I believe he talks about this in part 3 of About the Values of Chess Pieces.
I'm no expert on Chess, but I think about it this way:
It seems to me that in the opening position of FIDE Chess, most pieces are in quite poor positions, with their mobility greatly restricted by overcrowding and no immediate opportunities to threaten enemy pieces. An extra move can be used to develop your pieces into more advantageous positions, which translates into a higher probability of winning. As a corollary, the value of an extra move changes throughout the game based on how rapidly your position can currently be improved.
In On Numbers and Games, John Conway develops a theory of combinatorial games (Chess doesn't quite fit in this set, but it's close) as being a superset of numbers. Certain games (or sub-positions within larger games) have an exact numerical value because their existence gives a direct advantage to one player (positive) or the other (negative). But other positions have a "fuzzy" value; it's not clear how much advantage they give because it depends on how many moves each player takes (and in what order) trying to improve that particular sub-position rather than some other sub-position elsewhere in the game. He has a concept of "heat" that corresponds to the volatility of a position; the "hotter" a game is, the more advantage can be gained by whoever makes the next move.
(Note: I don't quite follow all of the math in this book and it's possible my description of his theory isn't entirely accurate.)
As for Betza's "quantum of advantage", I think his position was a little more complicated than that. He said that a third of a pawn was roughly the smallest advantage that a master player would notice in practice, and therefore that it makes a good approximation for lots of different minor advantages that are NOT actually equal but all roughly on that scale, and lists a tempo as one of several examples. I believe he talks about this in part 3 of About the Values of Chess Pieces.