I've been to places that offer tickets. Pay one price is so common now that I think people forget that tickets have their benefits too.
1. Non-riders: Before pay one price, amusement parks usually had free or cheap entry and they would purchase ride tickets if they were interested in that. The result was that amusement parks were truly *parks* where people could just come and relax or be with their families. Older parks often focused on dancing, picnicking, scenery, etc. in older times before amusement parks became all about rides. With pay one price, whether Grandma rides no attractions or 100, she's still paying $100 to get into the park. Pay one price has cemented the focus entirely on the rides and in the process "Amusement" in the form of attractions has come to dominate.
2. Pace: People using tickets are much more relaxed. If you have $20 worth of tickets and you won't be buying anymore, you have no incentive to rush because once your tickets are spent, you're done. That might sound unappealing to many people now, but it slows you down and makes the experience more relaxed. Now with pay one price many people run around like madmen trying to get "their money's worth." If you only have so many rides, you choose what you do carefully- which often results in shorter lines for everyone. If you can ride everything, you're more inclined to just jump in any line nearby if you get bored- as will several other people.
3. Capacity: Parks that use tickets have more incentive to increase capacity; the faster riders use their tickets, the faster they will buy more and the park will make more money (As an added bonus, if people have unused tickets, that is an easy way to encourage them to come back!) Under pay one price there is not nearly as much incentive to pay attention to capacity and operations, because the park already has your money when you walk through the gate. Notice that most of Disney's attractions that can really move people were built back when they were using tickets.
4. Ride Height/Interest Requirements: Too small or too big for an attraction? Don't care for gentle rides or thrill rides? With tickets, you don't have to pay for something that you can't or won't use.
Don't get me wrong, I will choose the pay one price option every time if given the choice, but it's important remember that tickets weren't around for so long for no reason.