I too was asked to provide my user (admin) password when getting my MacBook's screen replaced at an Apple Store. I had the same concerns as OP, but I asked if I could set up a new, separate user account with standard permissions (just for them) so that they could verify their work that way.

That was a fine compromise for them, but it did make me wonder how many other, less-savvy customers were providing their main admin passwords in this manner...