Yes those waxless gaskets are handy! Just make sure it doesn’t make your toilet sit too high off the ground or wax ring would be best. But I have been to many repairs where the toilet was set wrong and smashed the ring 🤦🏽‍♂️ so good point!

I’m a licensed plumber. If you can get ahold of the snake again undo the bolts and pull the toilet and try snaking through flange in the ground and before putting the toilet back you need to re install a new wax ring (very low cost) just so you don’t have leaks under the commode. If that doesn’t work try to find your cleanouts outside your house and snake through the one furthest from your house that one in most homes should be going inside the main under your house. All else fails that hole that was cut and never covered is going to be a major problem and roots will make way towards that feeding supply for them and also dirt and debris will catch on a hole in the pipe causing build up so that needs repair. At a hardware store they should sale what’s called a CT or rubber boot of the size of pipe that the hole is in and you have to just cut the section out and use two of those cts to re attach a new piece of pipe. If you have any questions contact me through pm if possible.