Is there a way to allow a senior employee or middle-manager to assign new accounts, to existing privilege sets, without granting that person full access privileges?

You can allow him/her to run a script which has the "run script with full access privileges" box checked.

Be sure that the targeted Privilege Set for the new account has exactly the privileges that you want to confer. Additionally, if you're doing this with more than one file, you'd be best advised to employ external server authentication.

Steven

What Steven said.

You can use the FM provide account script steps. Or you can remove the issue altogether by carefully crafting AND testing a privilege set that you can assign to an AD or OD group and those admins can assign people to that group without having to change anything in FM.