Toilet leaking around flange

To answer the question you asked, yes, you can get a new toilet that flushes faster. They have a flush strength label on the box, and the faster flushes are done by making the opening between the tank and base larger.

However, a slow drain may be more related to an improper vent or a drain line that's too small or improperly sloped. Without a proper vent, the flow down the drain is held back by the vacuum created by the trap built into the toilet. If the drain is too small, well, large things in a small drain take more effort. And if it's not properly sloped, we all know the direction this stuff flows, and it's not uphill.

To determine if the problem is with your toilet or with the drain, try pouring water into the toilet directly with a bucket and see if it drains fast enough. If that's better, replace the toilet. But if it's still not good enough, you've got some work to do.

it is possible that the wax ring is damaged, when another appliance like a washing machine drains the water backs up and causes the water on the floor. If this is the case it will probably get worse and your lines will need to be cleaned tree roots cause this very often in older homes.