Yes, as it would be a public good to everyone to be able to know where the potholes(that aren't profitable to fix for these private companies apparently) are so they can avoid them.

They might take a step back and realize that it would be more cost-effective to just own the roads, in which case your thought experiment ends where we are, because where we are was a place reasoned to(to an extent).

Pot holes do not enable fraud, ransom schemes, data breaches, denial of essential services to millions of people, and so on.

Doesn't the government use those software (private and open source) to handle private information of citizens and other sensitive information? And what about their contractors? That alone justifies maintaining such a database.