Your machine has a 15 AMP charging system, so at high speed, you should have sufficient current. Headlights & fuel solenoid will use around 5-6 AMPs and PTO probably around 3-4? HOWEVER- IF the battery is low in charge, it may "suck" a higher than normal amount and thus drop the voltage to the point that the PTO disengages. I would start off by fully charging the battery and see if that resolves the problem (at least temporarily). IF the battery is 2 or more years old, I'd definitely get it load tested. As a matter of course, you should ensure that the connections to the PTO SWITCH and PTO are good. You should unplug/plug a couple times to "wipe off" any "crud".

You can set your park brake, put your finger on the solenoid and turn the switch on. You should feel/ hear it click. If not, check for 12 v to the solenoid. You could remove the solenoid, slip off the loose parts, ground it and watch it work the same. Some of these have a little ground strap that fastens to one of the small screws in the mix cover on top the carb. Do you see the part of the plunger that would seat against the hole up in the stem? On many of these, if not all these style carbs, the little plunger will slip right out. I wonder if that is stuck up inside the stem of the carb? You could also remove the solenoid plunger and you could reinstall it to see if that was your issue.

***"I was able to push it into the garage"*** Did you have to select neutral in order to manually push the tractor to the garage? Or....can you push the tractor no matter what gear you select? It is important to know this because: If you can push the tractor manually no matter what gear the transaxle in in, there is likely some serious damage inside the trans. But, if you must select neutral position in order to push the tractor, the trouble is likely not inside the trans at all, but with one of the idler pulleys in the ground drive belt set up. The detail of "burning odor" is significant here because it implies that the drive belt is burning in the engine pulley groove due to another pulley being locked up. At least we hope that is what's going on. If the answer to the pushmobile question is "Neutral only", you will need to do a "hand on" check of both idler pulleys to see if one is seized (if so my bet is on it being the v-grooved idler pulley). Please post your answer and we will proceed from there.

In looking at the wiring diagrams for both the machines (917.276020 & 917.276310), I see no differences between the two except for the Kohler version (020) featuring an operator presence relay where the Briggs version (310) features a more "complex" seat switch to accomplish the same goal. If,........you are connecting the white wire to the small terminal on the starter solenoid, you are making the correct connection,........AND the solenoid is in fact mounted to a grounded, metal surface or has been provided with a separate grounding jumper wire to ground the solenoid case! So, you might want to test that the solenoid is grounded by touching the probe of a test light (or volt meter) to the battery positive post and the other end/lead to the metal case of the solenoid body. If you detect voltage or the light burns, the solenoid is grounded. If you don't detect voltage, it is not grounded and you need to address that first. So let's go through some checks to determine what the trouble might be. The following outline is a step by step that will guide you through. Refer to the wiring diagram for your 917.276020 for the identifying "alpha characters" I refer to in the text. Make certain there is not a blown fuse before you do these tests. (begin test outline) It could be a "bad key switch" but my feeling is "bad clutch/brake switch", but it could be a "bad PTO switch" just as easy, so let's do some testing first. You have a volt meter and/or 12V test light? Be advised that moving or wiggling wires while doing these tests could result in continuity being accidently restored AND this might cause the starter to engage when you don't expect it to. To guard against this, unplug the small white wire from the starter solenoid and cover the wire terminal with tape to prevent a short. First off, you will need to access the backside of the key switch. Step 1. Find the WHITE wire on the back of key switch connector. (leave the connector attached to the switch) While holding the key in the start position, probe the area where the white wire enters the connector body with a test light or a volt meter. You should detect voltage on the test light or the volt meter. If you do not detect voltage, replace the key switch. If you detect voltage, go to step 2. Step 2. Find the PTO switch and identify terminals C & G on the switch. Probe terminal C while the key is in start position. If you detect voltage at C, probe terminal G next. If you detect voltage at C but not at G, replace the PTO switch (make double dang sure the PTO switch is in the OFF position). If you detect voltage at C & G, go to step 3. Step 3. Locate the clutch/brake interlock switch (should be near the pedal or close by, must be attached so pedal linkage can operate the switch. This switch will feature 4 wires, 2 will be white. With the key in the start position, and the pedal locked or held down, probe the white wires of the brake/clutch switch. If you detect voltage on one white wire only, make certain that the switch plunger is actually moving in far enough to operate the switch. If the plunger is moving far enough to operate the switch, but you still detect voltage on just 1 white wire of the switch, replace the clutch/brake switch. The seat switch does not inhibit cranking on this mower, it only functions to kill the ignition spark. Malfunctions in the PTO switch or the clutch/brake switch are capable of killing spark AND inhibiting the starter. Remember to reconnect the white wire to the starter solenoid after testing. I hope this helps in troubleshooting the interlock circuits. (end outline) After doing these tests, post back your results.