Morris Minor Owners Club

Checked the dashpot level?
Also the plunger inside may be gammy round the edges.

Remove the air filter and you should be able to reach
the piston with your finger. See if you can raise it
without any binding and that when you release it it
falls down with a soft 'plop' as it hits the bottom.
When you raise it you should feel the resistance caused
by the oil damper in the piston chamber but no actual
scraping or binding. If the piston does not fall with a
plop then pull the choke right out and try again. If it now
falls with a plop then the needle is binding and you need
to go through the jet centering proceedure after checking
that the needle is not bent. If moving the choke makes
no difference then the piston is probably binding in its
chamber, clean it but do NOT use abrasives to do this. The needle could have been
moved too far up the piston,(the shoulder of the needle
should be level with the bottom of the piston). If the
mixture adjusting nut is right UP and the mixture is still
too weak then either the jet is lower than it should be
or the needle is too high in the tickover position.
One question,since yours is a 1971 do you have the
mushroom type breather fitting on your inlet manifold?
If so, have you started producing smoke from your
exhaust??

Also check that your float chamber valve is not showing
signs of flooding,( is there fuel present on the top,or fuel
stains down the bowl)?

It may well be the throttle slide sticking, or a jet problem, a float /fuel level problem as described already or a broken mechanism on the choke.
You say the choke is 'closing properly' - did you check it by pushing / pulling the cable so you could see the jet holder moving up and down?
You can also check the function of the throttle slide when the engine is running (presuming it plops down sharply as already described). Watch through the carb with the filter removed to see the throttle slide lifting (slowly!!) when you open the throttle and retuning quickly when released.
Apart from the unevern idle, and not needing much choke, does the engine seem to run fine?
Electrical faults are another (rare) cause of smoke but are unlikely to just cause a problem at idle.

I was wondering if it might have been something connected to that

Have you done a compression test? There has been a lot of talk recently of iffy head gaskets.

If all the above items check out ok then it would seem a
waste of effort to fit your spare carburettor. This looks like
a perfect time to use a Gunsons Colourtune which is,in effect,
a glass bodied spark plug which enables you to view the
actual explosion in each cylinder. the colour tells you what
the mixture is(orange=rich,light blue =normal) and it could
pinpoint a fault in one particular cylinder,or an overall fault
in all cylinders. I bought mine,long ago, in Halfords. I assume
they still sell it?

If the engine runs normally it's unikely to be the electrics causing a problem at idle. I had a car with a broken HT lead but it was very low on power, terrible mpg and a rough idle.
If the power seems normal and it's only rough at idle, then I second the idea of a compression check to rule out the pistons/rings/valves/valve seats/ head gasket from the possibilites.