I know this is meant to be a joke, but there are official sites in Ireland that really does use queues like this.
Whcih ones and why? Wierd load balancing? Or dumb bureaucracy?
Ticket sales events etc. often have them. Here's a company in the space https://queue-it.com/online-queuing-system/
Virtual Queueing / Online Queueing are the terms to find them.
The reason companies use them is for stuff like tickets releasing for an event at time X. People will spam the site and a random number of them will get through which incentivizes people to Sybil whack the site. Much better to provide first-come first-serve with no advantage to late joiners. Of course still not optimal, so some events will offer an initial queue to people who've been before or so on.