My sage advice is do not split that 22 just yet, you might open a Pandorras Box that will take potentially years to close again, particularly if you are not particularly mechanical, or have lots of tools, so try all the simple fixes first. It sounds to me like your clutch is working, the clutch brake is towards the back of the clutch compartment, just get a flash light and you will see it, and with nimble fingers you will get the adjusting bolt undone, and the clutch brake ring rotated into a position slightly closer to the main clutch, this will make a huge difference to its performance, and is not difficult to do, just use lots of loose juice.
I would not be scared to put a couple of gallons of diesel into that clutch compartment, and have it sloshing around while the flywheel is rotating, I've done that to my old Cats over the past 15 years. Caterpillar in their old Owners Books suggested washing the clutches with gas or kerosene if they became contaminated with oil, I prefer diesel, it is much safer, plus the oil content will help lube everything around the clutch, like the linkages, and plates, for better movement, just make sure the clutch compartment drain plug is fitted before you add the diesel.
Take that fuel tank off and clean the whole system, both the big tank, and the small one, because you will be surprised at how much rusty dusty crud you will get out of them, plus the 22 fuel tanks are well known for having rusted out bases, it was a design flaw in their support saddle, trapping moisture under the tank. Better you find out now, and get the tank repaired, before setting off to your back field for a days fun plowing, with 20 gallons of gas leaking all over your boots and a hot engine.......
I like to blow air or oil back through fuel lines from one end to the other, I'm assuming your fuel taps are working correctly, as is your fuel pump? If you are unsure about your fuel pump, test it, till then, just use the small pony tank. Next remove your carb and give it a really good clean, if your engine is trying to fire, that suggests to me you have some fuel getting into the cylinders, but not enough.
Your carb fuel bowl float valve might be choked with crud, or jets inside the carb are blocked, I'd expect to see that in a old Cat that has been sitting around for years. Modern gas is shocking on fuel systems, it will dissolve the zinc galv off a new tank, and will go gummy if left to dry out in carbs and fuel taps. As I said before, try all the easy fixes first, before even thinking about splitting that tractor, if it ran before, it will run again, even if it has to be tow started.
The most important thing you must do before trying to start that engine is confirm the oil pump is making sufficient pressure, this is critical. With every old Cat I've dragged home and I'm starting for the first time, after changing all the oils obviously, I remove the spark plugs so there is no chance of the engine firing off, and I slowly drag the old girl in gear and watch the oil pressure gauge closely.
If it jumps quickly up to 20 or 30 PSI, then I know I can move on to trying to start it, but if the oil pressure gauge doesn't move, or only shows a few pounds, you need to then check if your old oil pressure gauge is reading correctly, or start looking for why there is no oil pressure. Just dragging that 22 slowly, even hand cranking it with the spark plugs out, will quickly give you oil pressure if everything is ok.
The most important thing about this whole process is safety, it is absolutely critical you are 110% certain you know how to stop that engine if it does start, and that the fuel system does not leak. Too many good people have been killed or badly injured when a new to them antique tractor engine took off on them while it was in gear, and they discovered too late the ignition switch doesn't work!