It's quite rational because of the crucial importance of reputation internationally, and the image of competence domestically.

If by "internationally" you mean Western countries: Xi Jinping doing dumb things to save his "face" is very detrimental to his reputation there.

No one outside of China thinks his stubborn refusal to abandon the failed Zero Covid policy is in any way a good idea. Everyone sees it as a stupid and irrational face-saving action.