I always think as an individual I would like this. But at scale I worry it would incentivize each department to advertise themselves to the public, which seems to me like a waste of funds. I already dislike the reelection cycle (politicians incentivized to always be fundraising) and would hate to see that happen per department.

The police do something similar when a mayoral candidate promises to rein in police spending, it's called the blue flu and it's basically a strike.

Whatever hypothetical law allowed individual citizens to allocate funds could also ban departments from advertising or misrepresenting what they do.

Does PBS still do telethons like in days of old?

It would probably be wasteful initially but I doubt it would be more wasteful than bad policy that doesn't actually prioritize what the electorate wants.