Also keep in mind that with the some electric mechanical shift bendix type starters that due to the mechanical design the solenoid en-gauges/shifts the bendix pinion gear into the flywheel slightly BEFORE the starter motor starts spinning (slightly before the solenoid contacts make to spin the...
That sounds like it's might work ok because you are still using the starter's existing solenoid as the shift bendix gear mechanical and the add on solenoid's CONTACTs in parallel with the starter's solenoid contacts.
Make sure you do not use a wimpy China type add-on solenoid. The reason i say...
Just curious.
You say you REPLACED THE STARTER MOTOR and the solenoid contacts are not making good contact.
It's a solenoid shift type starter and the solenoid is on the starter housing.
I've seen the mechanical bendix shift linkages on those type starters get worn and act like bad solenoid and...
Appears you need to consider a replacement engine.
Several riding mowers used that type engine. (and salvage prices and or running [price would be less than cost of the parts you need.
You need to consider looking for a vertical shaft engine replacement.
I've got couple snapper's, a old Gravely...