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Currently, the next round of a tournament does not begin until all games in the previous round are finished. This does not have to be the case. Many times players are mathematically eliminated even before all the games are finished. Those games could continue for the eliminated players without holding up the players who are advancing. There's no reason why the next round couldn't go ahead and start once it's clear who the winners are. This would speed tournaments up quite a bit.
Tiebreaks are part of the formula too and could easily be accounted for when determining if players who are still in games have been mathematically eliminated. I'm sure chess.com would recieve complaints about why players can't continue but no more than they already do and perhaps less. I think the end result but would that the change in complaints received would be negligible and the speed of the tournaments would increase. A net positive for everyone.