You are right & the dealer is talking horse feathers.
OTOH putting a shut off valve in the fuel line before the first element is a good idea.
Shutting the fuel off and starving the mower is also a good idea.
The solenoid simply shoves a bung up the main jets rectum.
What the dealer was alluding to was a float valve failure which will allow the float bowl to overfill.
When this happens fuel passes through the bowl vent and then dribbles down the carb throat & into the cylinder .
From there it flows past the rings & fills the sump .
Now days the EPA mandates that no fuel can drip outside the engine.
So overflow & / or air vents all have to be internal.
A bit late for you now but usually you can clean the solenoid out.
They rarely fail electrically but crud accumulates around the plunger and jambs it.
Honda ones have removable plungers so you can pull it out & clean the hole with a Q-tip
Briggs & Kohler ones you dip them in your favourite cleaning fluid and worj the plunger in & out with your fingers till the plunger is free.
I use carb cleaner for field cleans &Innox for workshop cleans.
Some one on this list recommended popping them in an ultrasonic cleaner.
I have never done that before but will give it a try next time.
Withoug that plug you should throttle down then turn off the engine to avoid an afterfire in the muffler.
