If you can get the bike started and get the belt off while idling, you may be able to determine if it is just an engine tuning problem. If it still stalls with the belt off, you can't blame the clutch since there is no load. BE VERY CAREFUL. The tension of the belt against the pulley is enough to cut your finger to the bone or maybe remove part of it in some cases if your finger gets caught between the belt and pulley. If it runs fine with the belt off...or you don't think thats the problem then read on.
You may not need to take the variator completely apart to fix your problem. So you may not need the magnetic extractor tool. You can use the piston stop to remove the clutch nut, or an impact wrench works well (this is my preference) I never did have any luck wit the strap wrenches. You may be lucky enough to put a wrench in the flywheel side and break the clutch nut free. Worst case... you loosen the flywheel nut and retighten it (left hand thread). It is usually a buildup of rust on the the clutch bell that catches and stops your motor. It can also sometimes be due to a dent in the clutch bell that brings the pads too close. Simply pulling the front of the clutch assembly will usually do the trick. The clutches will be attached to the outside of that. You can rough them up with some 180 grip sandpaper to remove any glazing (shininess) This may help them engage quicker and stronger to "launch" you off the starting line as much as 50cc's can do. On the inside of the clutch bell, where the pads contact, use some 400 grit paper to remove any rust and dirt. A good hosing of the the whole assembly with brake cleaner will help prevent the dust buildup from causing future problems. Inspect the springs on the starter clutches. They can get really rusty and sometimes snap. You may also want to clean the areas inside the front clutch cover where the starter clutch pads contact. Again using 180 grit paper to remove glazing on the pads will help them engage. Put it all back together and give it a try.
If this does not work, it may be your caged bearing the the variator catching. You should always use plain translucent looking lithium grease on a moby. I have had zero luck with "bearing grease" and the Teflon infused lithium grease. the viscosity is too high for theses small machines... save it for the CV joints in the car. Taking the belt off and giving the variator a spin may not always tell you if its OK or not. If it grinds to a halt quickly the belt of, chances are the bearing is shot. If not, it may only have issues with the pressure of the belt on it which is harder to test. You may be able to disengage the mobymatic drive (put it in pedal mode) and spin the starter pulley with the belt on to see if its grabbing. This test does not rule out the bearing in your starter pulley as the source of the resistance though. If you feel you need to take the variator off completely to clean or replace the caged bearing, read on....
To completely remove the variator, you will need some long angled circlip pliers. The ones with interchangeable tips don't usually fit in there with the crank shaft in the way. A pair of needle nose pliers works well if you file the tips to fit the retaining clip holes. You will need this type of needle nose pliers.
Once you get front of the clutch assembly off, there may be a washer underneath. This is where the magnetic extractor comes in handy. Grease coated washers don't just slide out and good luck getting a screwdriver in there to flick it out. I have a cheapo Japanese multi screwdriver tool with an extendible magnetic antenna-like shaft hidden below the bit. This works great. If there is too much grease holding it in still, some brake cleaner works nice.
Next is the first of two circlip retaining clips. This one has given me some trouble in the past. The are more washers between this retaining clip and the next. The top retaining clip has a tendency to wear into the washer and so when you try to squeeze it to get it out, it gets caught up and won't budge. I bent my homemade retaining clip pliers a number of times before figuring this one out. Just use a small screwdriver to push down on the washer beneath the clip and maybe even wedge something between the crank case and variator assembly so it is not resting on the clip anymore. Once you get this clip out, use the magnet to remove the washers. Beneath this is another retaining clip. This one usually comes out easier. The variator should now come off the crank shaft.
At this point you can clean or replace your caged bearing. you may want to re-grease the rest of the variator while you are at it. The steel balls and central shaft usually need a good greasing to have it down shift correctly and cease the rattling at top speed. I don't think that the grease fitting on the end of the crank shaft takes care of this part of the variator, just the caged bearing. You can also do this on the bike by removing the belt, and sliding the variator cone forward and shoving some plain lithium grease in there. Push the cone back to the outside and wipe the central shaft before reinstalling the belt or you will have problems with slipping.
You should rare have to take the rest of the variator apart unless there are major problems with the variator cone getting stuck. If the variator balls are very worn or rusty, most of the time this just make it a bit noisy, but it still functions. Taking it apart the rest of the way requires some special tools as not to damage the variator. The variator tool can be fabricated pretty easily with a piece of plate steel, a drill, tap and some screws. A dremel helps too. That is a whole other discussion though.
Hope this helps,