Totally agree. I have rebuilt most types of carb over the years and it certainly is not rocket science (or heart surgery). There are very few parts of a carb-fed petrol engine that require 'specialist' attention (in fact I can't think of one) and the reason garages use specialists is time - specialists have all the parts to hand and can return a device looking like new. It certainly should not be because their mechanics are incapable of stripping, finding any faults, and rebuilding a carburettor. I'm concerned that the assumption seems to be that because you have sparks that are big enough and occurring at roughly the right time then the carb is at fault. Dismantle the thing, reassemble it as was done to the ignition system & the engine will run on the basis of that assumption. If it doesn't (and it may not) you have at least eliminated the simple stuff.
It would appear you have a flat spot just off idle that suggests weak progression (normally fed from the badly-named 'idle' jet). The 'idle' jet is the best fuel filter in the system because it has the smallest 'ole through it. Partially blocked it can kill progression but idle can be adjusted (idle mixture screw) to compensate. the symptoms would be as you describe.
A poor supply to the ignition coil (one that looks perfect viewed with a multimeter) can occasionally do the deed. Temporarily run a wire from battery positive to ignition coil positive to check.