I think one of the complaints around efficiency or rather inefficiency is when someone has an ailment and needs multiple appointments but with different people / departments but they have to go for say 3 appointments on 3 consecutive days, why can't they all be scheduled for the same day even if not 1 after the other

But is it actually any better in the private sector, i'd say not from my experience -
A few years ago my Mum needed an operation and decided to go private as she basically couldn't get any progress or date from the NHS, her appointments were about 45 miles away which meant a family member freeing up time to take her and none of us lived near her, multiple appointments none of which were on the same day even after we asked if things could be scheduled accordingly.
The same with my dentist were the last time I attended about 5 months ago I had another check up and dental hygienist appointment booked for a few weeks time back to back which makes sense.
A few weeks after making the appointments I was notified that one of them is now the following day as someone has apparently left, so why couldn't they simply transfer all the existing appointments over to the new person

This really fucks people up if they need to take time off work etc

It's not just a one off either as the last time I was in there another old bloke was complaining about how he now had to come 2 days in a row and had made the appointments back to back to avoid that as he needed a hip replacement and struggled with the stairs