Yes. All the minions you see are limited in number. Lower tier minions having more copies available than higher tier minions. So as the game progresses, you will see fewer copies as your opponents also have access to the same minion pool.
Random is random, but tbh sometimes it does just seem like what you're looking for is just not there. The amount of times I've gotten to T6 early and refresh constantly over 5-10 rounds and you don't find it even if there's no competition from others. I mean i've had it happen before where I'm the only one playing pirates, i have infinite gold going with hoggarr's and refresh literally hundreds of times and make tens of goldens and zero Eliza's appear.