The Vanguard engines have magneto ignition system. No 12V input for ignition. Has this machine been out of use for a long time (year or more)? To determine for sure if you "have no spark", you can use a "known to be good" spark plug for testing. Open the gap on the good test plug to about 1/8" so you can see the spark easier. Connect this good plug to one of the plug wires. and making certain the metal body of the plug (part that includes the threads) is touching a metal engine surface. Position the gap so it is visible to you. Crank the engine while watching the spark gap. If the engine is making a spark you will see it jump the gap. If you do not have spark, and this machine has been unused for a long time, enough rust may have formed on the surfaces of the magneto coils and the rotating magnets to reduce or prevent a spark being generated. Another possibility (but this is a long shot) would be that this engine has the optional (Briggs & Stratton option) "low oil level" protection system in place. It is an available option, but I have never seen it used except in stationary equipment where the engine typically runs without any operator supervision. If Simplicity opted to put this on a machine, there should be an LED light somewhere (on the dash I would hope) labeled "Low oil". So far as I know, there is no other "spark kill" circuit that would prevent the magneto from sparking on the Vanguard engine. All Vanguards have an "Oil Pressure" switch located near the oil filter that lights the "Low Oil Pressure" light on the dash. This IS NOT the low oil LEVEL shutdown I mentioned. If it has the oil shutdown there will be a sensor somewhere else on the lower portion of the engine. My understanding is that all "engine cutoff devices" , including the ignition switch and any "safety" interlocks, affect that by interrupting current to the fuel shutoff solenoid. I have no Simplicity experience but some other forum members do. Hopefully one or more of them will shed some light on this subject. Meanwhile, make certain whether you do or don't have spark. If you have spark, it's time to begin looking at the fuel system.

Update....the engine is running!!! My son and I had faced a setback earlier in the week when one of the head bolts sheared after the second round of torquing to 36 ft-lbs. That was a bummer, but a couple days later we got the bolt out with an easyout. We put all the parts of the rider back together (with a new needle/beam torque wrench) and fired it up with some carb cleaner, and it started up fairly quickly!!! As expected, we did have the smoking of burning off of oil, but after 20 minutes or so, that seem to diminish to nothing. Seem to have a little bit of burning oil smell still, but I am hoping that is just some residual oil burning off the frame or muffler shields - those parts around the muffler seem to get very hot. Only remaining issue was a backfire after turning it off. It happened half the time after shutting it down, even when I would let it idle for 15 seconds before turning it off. But I can probably live with the backfiring. I would welcome any input on this problem, but I can do some research on the internet about this seemingly separate issue. Thanks so much for walking my son and I through our first fix of this magnitude....usually we only fix fuel problems. It was a really satisfying experience, and I am really grateful for your advice along the way....I was truly amazed. Again, thanks so much.

Let's concentrate for now on the finding that "one of the valves does not compress as far as the other." That is significant, especially if you can tell the difference without any measuring intruments. Valve adjustment is a precisely orchestrated series of steps to get the engine positioned to the right point in the 4 stroke cycle before you can check or adjust the valves. Now, let's see what we can find out about "why one valve doesn't compress as much as the other". You need to get the engine to Top Dead Center on the COMPRESSION stroke. Because you have little or no compression you will not be able to sense it by placing a finger over the spark plug hole. Instead you will need to watch the valves and rocker arms as you turn the engine by hand. Rotate the engine until BOTH rocker arms have no contact with the valve stems. When you reach this point, take a small pocket ruler and measure the height of each valve spring. Measure from where the bottom of the spring rests on the cylinder head up to the edge of the cup washer on top of the valve spring. The springs should measure exactly the same for both valves. If you find one spring appearing shorter than the other, you will also find it is the exhaust valve. This indicates that the valve seat has ejected from the head and is holding the valve open. Head will have to come off to repair this (if this turns out to be the case). Check the springs length and post back.

You're on the right track looking for the reverse position switch. Those are the classic symptoms of a mower that "thinks" it's in reverse. It won't be the key activation switch. I suspect that the switch is activated by the reverse pedal position, and is probably jammed into the actuated position by something, dried grass, or a broken twig, or maybe just a bent actuator arm, etc. steve