I know this is thread is a year dead, but I wanted to add my experience for anyone who searches for it.
I've got a saw rebuilt from a smashed ms261c and the carcass of an ms261. I couldn't get the Mtronic carb/ignition working so I've been running it as a ms261 with a adjustable carb and regular ignition coil. I have/had all the key Mtronic parts: flywheel 1141 400 1230A, ignition 1141 4701 A (1634), and the Mtronic carb and wiring hardness, so several times I tried to convert it to a ms261c. Every time had similar issues: sometimes it would fire once/twice but never stayed running even for a second. I would yank and yank and get the cyl/muff warm from firing but never got it running. I tried everything I could think of short of buying coil/carb/flywheel: replaced the spark lead and boot, tore down the carb, adjusted the air gap, triple checked the wiring, tried the non mtronic flywheel (I was desperate). Read a ton on here and and other forums and a number of people blamed ignition coil, but its very expensive.
Recently I was determined to fix it and paid $50 for a mtronic coil off fleabay: still same issue, with the spark plug out of cyl I could see two sparks when the angular velocity of the flywheel increased sufficiently, but never more than that.
I finally read this thread carefully and got the idea to check wiring harness connection. Sure enough, yellow wire from module to carb was open circuit, seemed to have been broken where it inserts into tight slot in crankcase. I crimped/soldered a new wire on and she fired right up, one pull choke on, one pull choke off. Thank you to the folks above who suggested this.