Having worked as a controller at an airport that was favoured by several instrument training organisations I can maybe explain the simple reason for pre-booking.
Operating a procedural approach service with one beacon and routine approaches to a go-around is hard work (and can involve a lot of R/T). And I don't mean hard for the *poor* controller. The rules about where aircraft can go and how far apart the must be can be complex and, in practical terms, it's not normally possible to accommodate more than two aircraft at the same time. Add in a few commercial movements which the airport, quite reasonably, doesn't want delayed and two trainers can really make a mess of things!
If you are VMC and can see the other traffic that you're being separated from it might seem an awful long way away - but that's only because the rules are set that way.
If the controller has radar things get a bit easier because even if you are training and getting what sounds like a procedural service, the controller can watch where you are and bring other aircraft around you at much smaller separations than a procedural service requires.
It would be nice to suggest that you ask to visit a busy procedural approach unit to see how it works but there are so few around. But do try if you get the chance.
And one last thought. When you hear a controller talking about doing approach control with his/her eyes shut they're not bragging, it's just that that's what it feels like!
