Interesting differences. Here we have a private clinic too, it's free though. The forms are from them, and only have their addresses I believe. You make your own appointment, and can do it online or call the number. That's the lab (any of their locations) my hematologist didn't trust though, and I think they did screw up the test once as the value was much lower than the other 3 times. They also take walkins, but in the morning you'll usually have to wait a while as that's their busy time due to the fasting tests. I have walked in for a fasting test in the morning, but not for one where you don't need to fast. I've had issues where a location wouldn't do a test, central booking said they did do it and it was a bit of a mess as something was also time dependent after I had taken a bunch of pills the night before specific to the test and had a two hour window. I think that time, I needed to get the tests split into three, one done privately in the window period when I showed up, then one at the hospital that had to be done the same day, and then a fasting one even thought the form didn't ask for a fast.
The hospital lab is walk in only. It used to be a long wait, but I think in patient samples were sent there too. Since they opened up the clinic across the street that's all out patient there is next to no wait for the out patient lab. Unfortunately, at the hospital there is no free parking, and the private labs usually do. If you get through quickly enough though, you don't have to pay and they usually are that fast, even in the morning. I don't know if people just don't know about it, or why they aren't very busy.
In the news a while ago it was mentioned the private company's contract wasn't being renewed. I think another company was likely going to take over. The current company was going to try to fight that, not sure what's going on now though.