Comes on but cuts out and bellows smoke from filter.

Possible fuel pressure regulator allowing too much fuel pressure.

Try changing the diesel filter, when old and clogged they cause bellowing black smoke

the chimney (flue) is blocked
check for blockage at the outlet or in the pipe

WELL BLUE SMOKE IS A CARBON MOST LIKLY A VALVE BROKE AND IT WOULD JAM A MOTOR

First you need resolve the carburetor issue. With it leaking fuel it can flooding the engine causing it to smoke. The leaking can dilute the engine oil by overfilling the crankcase causing smoking and other damage, Once carburetor issue is resolve and it still smokes the initial 10 minute start-up after the rebuild then a compression and leak down will need to done to determine if you have blown head gasket, bad rings, or other problems.

Sounds like you may have some internal engine issues. Check your spark plug(s) to see if they are oil soaked. If so, you may have some issues with the engine.

I would start by checking for a fouled spark plug/wire. Also, check the fuel filter. It doesn't have to be good just because it is new. Could have a factory defect.

As u know - that smoke is oil . can be cause by restricted breather or oil filter, worn valve guide or seals. or a bad piston ring which seems the most likely if its just coming outa one pipe. A compression test will tell ya if thats it

the white smoke is oil - did you overfill the oil ? if you tipped the machine your air filter could be full also - if this is an old machine it could be ready to throw a rod but not enough info from you to tell

black smoke from exhaust means a rich oil/fuel ratio, sometimes it causes by a clogged air filter or a problem on fuel injection system.