Most teaching barns Ive boarded at that had Beginner groups did not accommodate late arrivals, especially riding School horses/Ponies. Often they run several beginner lessons back to back so it not only disrupts that lesson but the next one as well. Not fair to the other riders. Not that unusual to put another rider on that horse/pony if the scheduled rider is not there at the start time either, maybe an upgrade for the prompt rider, maybe to give another busy Pony a break.
Far as I know they were charged and did see some late riders just rolled to the next lesson if at the same level and there was something available to ride.
Far as checking the girth? I didn’t check mine once and I was no beginner. Seen others do it that were way past me. Human frailty. In my case it was a Western saddle. With breast collar buckled on. And a back cinch properly adjusted to mount. Nobody else there, thank the Lord that horse was a broke to death QH who wanted to help me, not kill me and kick the saddle to bits. Should have seen the look on his kind face as he turned his head to see me balancing that saddle on his side trying to shove it back up. Must have thought what an idiot, swear he heaved a big sigh of disgust and rolled his eyes.
Never, ever, not no way failed to check girth before mounting since that day. But I dont wish being reminded to check it that way on anybody else as a learning tool. Though many will do it at some point in their riding career. Even Pros when somebody brought them the horse tacked up and groom assumes rider will check and rider assumes its ready to go. Usually the only harm is to rider ego and the first thing they do is look around to see if anybody saw.
Have seen most, if not all, Clinicians at least eyeball each riders girth and a few actually check with a finger…although they are as likely to chide rider for too tight or where saddle is positioned.
So, no, I am not the least offended by girth checks for any rider because trainer cares not just because its a requirement.