Often, you aren’t able to choose the speed as a tank, though.
I am haunted by a memory, a while ago now, of when I was levelling a Monk. I didn’t like Monks much - still don’t - and decided to dungeon spam instead of taking my time.
Went tank for quick queue. I hit Deadmines. There were these 2 Enhancement Shamans, I suspect they were friends, fully loomed and in no mood to wait. The healer was clearly a beginner who had no idea what was going on … well, it would have been clear to me if the two Enhs hadn’t decided to go full Rambo and start sprinting ahead of everything. I felt obliged to go grab aggro and get back to the front. So I did.
Yeah, complete beginners have no business signing up to heal, but complete beginners also have no way of knowing that they shouldn’t be signing up to heal.
Anyway, while I ran to follow my people so I could lead them, the beginner, and a DPS who very decently stayed with him, were getting into all kinds of trouble with random mobs we had left behind. I didn’t even realise I had left them behind until we were at the walkway to the ship.
Normally, when tanking, I like a moderate, steady pace, and have a priority of KEEP THE GROUP TOGETHER. Avoid skips, tricks, theatrics that may leave someone not understanding what I’m doing. But I was never going to keep the group together that day, and to be honest, I wasn’t even aware enough to try. That day I felt pressure to keep ahead of the terrible twins.
Ever since, I wish I had apologised to that beginner, and explained.
So no, while tanks can have some influence on the group’s speed, if they have enough experience, they can’t avoid rushing in all cases without causing other problems.