I removed it to eliminate the possibility of it stuck closed, not letting fuel come in. But that obviously isnt the problem since it wont even fire with starting fluid. I might as well put it back on.
At least you didn't cut the tip off like DYIers do, destroying a perfectly good solenoid.
Don't laugh folks either; new solenoids do fail to retract. I had three brand new Brigg's solenoid that fail to retract straight out of the package. My distributor refused to replace them say Briggs changed their policy and that I had to get a local delaer to warranty them. Well that doesn't work around here as all the I contacted refused to honor the warranty even with proof of purchase from the distributor because I didn't buy the part from them. So I am out over 150 usd for the parts.
The failure of the part was that it would not retract; unless, you bumped the pin. It was as if they 24 vdc versions instead of 12vdc. Anyways the customers mower and one my personal mower have been repaired with used solenoid recovered from blown engines.
And Chughes2009 Reinstalling it would be fine.
They are best removed by placing a pry bar under the flywheel and prying up then with your other hand give the loose nut / bolt a strong "tap"
Pullers have a bad habit of warping flywheels.
Personally I prefer using a harmonic balancer puller, tighten up. If flywheel is stubbornly stuck on the taper then using a brass 2 lb hammer smack the center forcing screw to shock the flywheel off, may need a few re-tightening of the center forcing screw. Anyways it is better than the possible crankcase or under the flywheel components damage. Some the newer engines have might thin crankcases and I have seen the pry bar method to break the crankcases.