It would probably get rid of all bidding period, and it would create an unnecessarily adversarial relationship between parties who need to work together productively across multiple projects.
Also, if you follow this logic through to it's conclusion, you end up with a bankrupt contractor, a project that's still unfinished, and embarrassed politicians. It's usually still a better deal (within reason) for the person already doing the work to be the one to finish the work.
It seems like you could include penalties if project goals are not met, and then open up a re-bidding process. If it's true (as I think it generally is) that it's "usually still a better deal [..] for the person already doing the work to be the one to finish the work" then they can presumably bid lower than the rest at that next phase - but it's not just free money.
Also, if you follow this logic through to it's conclusion, you end up with a bankrupt contractor, a project that's still unfinished, and embarrassed politicians. It's usually still a better deal (within reason) for the person already doing the work to be the one to finish the work.