I believe armageddon is fair only if they bid, but just because they bid doesn't mean it's fair, i.e. bidding is necessary but not sufficient to be fair. Which side has the advantage in an Armageddon tiebreak game?
Paraphrasing what Jim Ratliff says here:
It would be superior if the rule were, instead of (a) the winning (i.e., lower) bidder gets their own bid on their clock, that (b) the winning bidder receives the other player's bid on the winning bidder's clock. This would be a form of a second-price auction, where each player has the incentive to bid their true value. Otherwise, each player A has an incentive to try to outguess their opponent B and just barely undercut B's bid, even if in fact player A would have been willing to accept even less time in exchange for their colour of choice (usually black).
Consider recently in 2023 champions chess tour Airthings Masters magnus vs hikaru, where the bids were Hikaru Nakamura 8:59 and Magnus Carlsen 8:58.
If Magnus is just guessing Hikaru would pick 8:59 whereas Magnus would go as low as, say, 7:00, then Magnus has a 2 minute advantage. Unsurprisingly, Magnus won and did so specifically by draw odds.
So I guess on the surface it appears that Hikaru had a 7 minute time advantage over Magnus but really Magnus had a 2 minute advantage over, well, Magnus.