Guys I think several of you could teach a class on this but I thought while I'm in between things I'll offer a few things that we see regularly on the bench that needs attention on a no start 2 stroke.
A good long T27, T25, 8mm deep socket X 1/4" drive, hemostat and a few metric allen hex T handles will help.
Any doubt about compression, pull the muffler and take a peak at the piston, ring(s) and cyl bore condition. No need to revive a trashed cyl with new fuel system parts.
Old fuel. Any doubt, drain and refill. We drain it into a clear plastic large container and look for crud, clarity and water settling out. We've seen straight gas, wonder what that gas is, diesel and even one contaminated by refueling from a used Roundup container. That was nasty.
Purge or primer bulb if equipped. Cracked bulbs no worky. R&R. Note which line goes where on the purge fittings and the entire fuel system.
Fuel lines. These age out, crack, deteriorate and do anything other than transfer gas from A-B. A long pair of needle nose or those hemostats really help.
Fuel line grommets. These are the seal between tank and lines used on some models. Same as fuel line aging etc. Fuel filters while you're in there.
Carbs. sometimes its just economical to use an OE part number carb, sometimes not so. When we take a carb apart and check it we pay real attn to the metering diaphragm. It has the button on it that pushes against the fuel needle lever. When these age out they get "crunchy" yup they will make noise or snap as you flex them should be supple and really freely moving.
Carb kits. Yes you might get lucky with a kit. Walbro and Zama have them and sometimes a good shop can ID the kit by eyeballing the old carb sometimes by the numbers on the carb. I just did a Homelite for a family member and it was a Zama C1Q H45 as I remember. That can get you to the kit.
Carb gasket and diaphragm stacking order. Lay it out on the bench so you can easily see what went where and in which direction. It matters.
Recoil rope fraying, replace it!
Lots of the rest are basics.
OK now for some oh crap, its dead!
Any screeching noise as you pull the recoil with the plug out, pull the recoil and look at the flywheel magnets. See any scratch marks? Now CAREFULLY inspect the crank bearings for play. Crank movements from bad bearings that have allowed the magnet to scratch against the coil CAN be fatal if you're paying to have it fixed if any other conditions exist.
Cyl and piston scarred Evidence of oil weeping behind the flywheel or deep behind the clutch and a saw that can run on full choke but dies when the choke comes off. Likely the crank seals are leaking and allowing false air to enter and mess stuff up big time.
Vacuum leaks on the carb to cyl. Many have a rubber boot intake coupling, inspect for splitting or holes, again false air.
Back firing. Yup we had one that jumped timing due to a sheared off key in the FW. That one was repairable.
May more exist but that's a start.