From what I understand, wait time can certainly be an issue with single payer healthcare. However, there's people in the US who have effectively infinite wait time because they can't afford treatment at all.
I have an excellent insurance plan and ready access to a large US hospital system. The wait to see a dermatologist as a new patient is ~6 months. Definitely not unique to single-payer systems.
Also, this wait times in many part of the US are in line with the single payer countries. The quality of care in the US is heavily dependent on location.