@markes could be any of the safety switches as well as a bad ignition key. I know you said you did bypass the switches but the only way of doing that for sure is by applying jumpers or connecting the cables without the switches. There should also be a fuse that comes from the ignition switch to the solenoid, check on that as well. I am sure you already checked the battery wires and ignition switch wires for corrosion etc. and your new battery is fully charged. Use a set of booster cables and see if you can get the starter to work by using those cables instead of going through the wiring of your mower. I attached some wiring diagrams of typical wiring for a riding mower. Let us know what you find out and about the model etc.