The free-spool valve should be on top of the 'charge plate' section toward the rear between the 'relief' vlv caps that are just above hose outlets, if yours has it. Some of this stuff is a bear to get at. For that reason I hate to see hoses come off ... awfully easy to get grit into either major component afield & wreck more than one.
Don't know which engine you have. Mine's a Honda GX-620. Learned a lot when helping another member re-power his T5C. Turns out that my older ignition is 'ground to stop' and his GX-690 (newer series) is 'ground to run'. Know which yours has, including seeing an 'M' terminal on the ign switch with older power. My I-S had been replaced lacking that & I used a relay's NC terminal to provide the gd when switched off. (Not sure which way Kohler would be.)
The reason I mention this is because you may want to check that your coils are grounding
or not for whichever ign setup. Guess wrong and you'll be banging your head .. or not starting because of the possible confusion. (My 50" ZTR has more complicated wiring than my T5C.)
Your T-M may also have
two electric fuel shutoffs to check out. One is on the 'firewall' and the other would be on the fuel bowl of the carb. (I added a manual one, too) Note that the tank's outlet is just a fitting threaded onto a welded-on boss at the bottom. There's nothing to keep swarf or rust flakes from plugging it up, and I've had to huff air thru' to clear that more than once while on a dig. These guys are mostly easy to tinker with, but a bit of 'been there' sure helps with diagnosis.
edit: It would be nice if T-M's manual(s) (ie:2006) didn't mix the years/varieties as shown in the pix on ppg 2.4, 2.10, etc. Be assured that the lads who answer the phones will
not be as confused as we often are.