I'll always defer to the experts when it comes to these questions but a hand troweled finish makes me nervous. If you give some finishers an inch, they'll take a mile and you could end up with even more grinding.
There are two operations that occur with a power trowel, floating and finishing. Floating determines the localized flatness of the floor (think waviness) and is very important. The second step polishes the floor using steel trowel blades. This is the step you want to go easy on.
I'd want my finisher to float the floor with a power trowel, knock it down and trowel it flat but stop short of a polished finish. I'd call this a smooth troweled finish. The last step could be done by machine of by hand. The goal is to flatten the floor and remove any trowel marks but stop short of hardening and densifying the surface.
It can be a difficult concept to communicate because the terminology varies but it's awfully important to get it right.