hey there! just one perspective for you... at my old lesson barn, we had to arrive at least 30 minutes before our lesson was scheduled to begin to fully groom and tack up our horses. we had curry combs, brushes, hoof picks, etc. available for all the lesson horses that we could use for grooming. it was a great opportunity to just spend time with the horses and get a feel for their personalities (sometimes before riding a particular horse for the first time. i really enjoyed getting to do this, and i would often get to the barn ridiculously early to just hang around, watch trainers lunging horses, spend some time with my horse in her stall, etc.
in your case, though - even if you don't, for whatever reason, get to tack up your own horse, i don't see any reason why you wouldn't be able to arrive a bit early to hang around with the horses and spend some "quality time".
having to groom and tack up your own horse is also a great way to learn about how to handle horses with, shall we say, "different" ground manners, and such!
personally, i would hate it if i had arrived at my barn with the horses all ready to go - you're right to think you're missing out! but again, see if you can't sneak in some quality time when you're not riding; you'll definitely benefit from it.