Note that I said “admitting” — I personally do prefer becoming more correct over time but it's hardly a novel suggestion that most people do not enjoy having to retract a previous public position. This is everywhere in society from casual use of terms like “saving face” to work by psychologists, economists, etc. on how to support people making better choices rather than being ruled by the sunk cost of past decisions. This is why public retractions have such weight: even people who like being right don't enjoy that part of the process.
This isn't about treating communication as a contest with only one winner but rather about the subjective feeling of the person making these decisions. Having built it up into a big deal increased the costs of reversal but the mandate changes the dynamic and gives them the option of saying the choice was no longer theirs — if they get vaccinated, that's a better outcome for everyone even if they never admit the bad call.