They literally can't do that currently
Yes they can. (Thought we'd had this discussing in PMs some time ago, where you'd conceded this point.)
The club simply encourage people to log in and then be placed in the queue proper at any point during the hour before the sale starts. You manage that by a separate Queue-It pre-queue. People pretty quickly learn there is no point in having multiple sessions, so the number of people logging in = number of queue positions.
Getting everyone logged in and then into the queue proper between, say 07:15 and 08:15 should be a doddle really. It would be pretty similar to how the queue goes down lightning fast once a sale is sold out. Obviously, if you leave it late, like 08:10, before trying to log in, you are risk of not getting through the pre-queue to the login process, and then on to the queue proper, but you'd presumably only make that mistake once.
Of course, a system that gives every member just one entry in the Queue-It lottery might be a scary for some. But no-one could argue about its fairness.