What you say is true. What you're neglecting is that you need a random process to choose who will go first. Let's use your own example: If four players go around the board 100 times, there's a near 25% chance that a given player gets around first. As you correctly say (indirectly), you will asymptotically approach a 25% chance as you increase the number of rounds towards infinity.

What you seem to be forgetting is that there's a very easy way to skip the infinite number of rounds, and get directly to the 25% chance: By choosing randomly who goes first. Of course, you need to do that anyway in order to start the warm-up rounds at all, so what you are effectively doing is

First: Give every player a 25 % chance to start. Then: Spend an arbitrary amount of "warm-up" rounds to randomly choose a different player that gets to start the real game.

Of course, these are not independent random processes, so the player that wins the first selection has an advantage in the second selection. The overall probability that a given player starts the "real" game first then becomes identical to the probability that they start the "warm-up" first. An infinite number of warmup rounds is literally identical to a single dice roll in terms of the probability that a given player goes first. So what you're doing is one quick random selection, which you immediately throw out in favour of an infinitely time consuming random selection with the same distribution.