Would a Project Manager be able to do this?

[Hotdogman1]

Yes! I am thinking a big missing piece of the puzzle is the scheduling software. And the “right” people need to take charge of the spreadsheet.

Too many staff, too many sites, it'll have to be split into multiple spreadsheets (one per site or 1 per X sites), multiple staff with the same role.  There's probably a 'go-to-number' for the staffs, but it can't be just like 5 people or so, hmm...

I think you are underestimating how inefficiently a department can be organized. In my department alone, I have 15 people who outrank me with 10 having the same exact job title. I share my rank with 3 other people and I outrank 15 other people who have the same exact job title. I'm the only one that has a unique job title other than the director.