Ride-On Mower only cranking once when starting

I have a Husqvarna TS 242 with a Briggs and Stratton 40N877-0016-G1 engine that I’m having some trouble with. My suspicion is something with the electricals – here’s what I’m seeing.

After a few weeks of wet weather, I went to start the mower, and the battery was drained, and would accept a charge. I replaced the battery and now when I hit the starter, it only cranks once. With some starter spray, I can get it to turn over, but it doesn’t run.

To further elaborate on "cranking only once" – when I press (or hold) the starter button, the engine will complete only a single revolution. If I have some starter spray in the cylinder it will turn over and run for a few seconds. Without starter spray, nothing – just one rev and that’s it.

I replaced the spark plugs and if I keep spraying ether into the air intake, it will run (smoothly), indicating that compression and plugs are okay. (correct me if I’m wrong)

From there I assumed fuel issue. However, none of the fuel lines are blocked – if I crank the engine I can catch fuel out of the pump-to-carby line, and I’ve replaced the fuel filter. The fuel in the tank is fresh, and I can’t find any occlusions or blockages anywhere.

Now I’m thinking solenoid – coupled with the fact that the starter doesn’t crank continuously – it feels like something is off with the electrics. The mower is a push button start with a security code. This functions as expected. The starter does not function if the blades are engaged, park brake is released or the seat is unoccupied, as expected.

So my questions are:

• Is a starter cranking only once a common issue with an easily diagnosable fault?
• If not, what should I be checking?
• Are there any model specific things I should be aware of that make this difficult to remotely diagnose?