I guess in a cynical way it very well might, as it fits the bureaucratic and political mold perfectly.

Businesses would be able to rubber stamp a "Verification of Correctness", and the government could parade this political achievement around to people not knowing better, satiating their hunger for better quality software (supposedly).

In the meantime, programs would indeed feel like they became better. Except that'd be less due to them being formally verified, and more because generating all the formal verification artifacts would practically require using agents, and those agents would incidentally produce better work than what's currently typical. Not the least because people would more readily pose tough requirements to them, without regard for the difficulty.

Eventually we'd then get back where we started, with programs being flawed, just flawed in a consistent way from some arbitrary perspective (so as to still pass formal verification, of course). Since regulation would be obsessed with the rubber stamp rather than anything else, the businesses would continue to float about as usual. The only thing that'd change would be the nature of the issues.