But a slower rotor also stalls quicker, so it's a balance (not really a concern in a fan jet design).
A slower tip speed propeller loses its propulsive power faster.
It is possible to design a turboprop which is pretty fast. Tu-95 has swept wing, counterrotating turboprops - and it is very fast compared to ATR, or even Piaggio Avanti. But Tu-95 is extremely loud as well. How does Tu-95 sound compared to the jet B-52? And how does fuel efficiency compare?
Tu-114 is the fastest civil propeller plane, too.
And at the other end, it is possible to design a turboprop with a very large diametre, slow tip speed propeller. Sure, ground clearance is an issue when you have a low wing, like Tu-114 with its tall landing gear, but you can go to high wing, gullwing, engine pylons above wing...
Interestingly, the abandoned propfan concept where the blades were unshrouded did reduce fuel consumption by 35% (compared to 90's turbofans, that is), but it did produce noise problems.
How does Piaggio Avanti compare to jets of similar size?