The wall of text is great! It’s refreshing to read a detailed report about a problem, instead of “my car squeaks” and having to ask 20 questions to get some relevant details about the conditions in which the problem occurs!
Can you see any fluid leaks? Why would you replace a cylinder if it is not leaking? A good learning experience, but I doubt it would solve your problem.
Honestly, I don’t see how bleeding the clutch could help, either, because if your clutch needed bleeding, then it wouldn’t work in the first place and you’d have real trouble getting it into any gear/changing gear when the engine is running. Maybe your described double clutching is a symptom of air in the system but I doubt it (says synchros to me). If you have trust in your mechanic then you would think he would have bled it during clutch replacement, or maybe you could ask him if he did? You could get air in the system if you actuated the clutch pedal when you didn’t have any fluid in the reservoir, or it could be because of a leaking cylinder, otherwise I wouldn’t describe it as “easy” to get air in the system.
If I were you, I would do a few different tests, and find as many tests as possible to get a better idea. Here are some:
- Start the car, select 3rd gear, slowly SLOWLY let out the clutch, handbrake still on, no throttle. The engine should start to bog and then stall when the pedal is past biting point if your clutch is good. Then again, with MX5’s you can do that in the lower gears, too, but 3rd is the classic test gear.
- Stationary, car on. Plant right foot firmly on service brake (just in case) Clutch pedal depressed throughout, select 5th, then 3rd, then 1st, then 4th, then 2nd etc etc etc, go through every gear change from each one to each one, making an effort to make gear changes you would not normally make and take a note of any that are difficult to engage. What you are looking for here is the condition of the synchros in the gearbox. Reverse not so important here but can be worth trying.
- Find a gear that is hard to engage (and also try it with reverse), and keep pushing the gearstick into it as though you want to change into it. SLOWLY raise the clutch pedal, does the gear suddenly engage? This points towards synchros again. Or does pumping the clutch pedal all the way up and down help? This points more towards clutch, possibly air in the system.
- Take your gear shifter apart and look for play in the selector. This might tell you if it is a selector issue.
In general, if the problems are intermittent, it is likely to be clutch, if they are constant, it is likely to be the gearbox.
In terms of flywheel, I think people put way too much stake on the flywheel on these little cars. Unless it has been absolutely thrashed to all hell and is incredibly (and I mean incredibly) worn/warped as a result, the flywheel should be of no concern. The clutch is, of, course, designed to wear WAY before the flywheel does and it takes a monumental amount of abuse to get a flywheel into the sort of condition where it will noticeably affect the running of the car. Your trusted mechanic would have checked the flywheel during clutch replacement, and pointed it out? If the problem was flywheel you would have thumps and other symptoms in every gear, not just first. Although if the thumping-shuddering continues throughout the gear after the clutch is released, that points to problems in the gearbox and could be anything from synchro-rings to broken teeth, impossible to say without taking the gearbox apart sometimes.
As for the shudder, is it a result of the revs dropping and engine struggling, or do revs stay the same? If revs dropping, this is simply a skill issue (you drive many manuals?) If revs stay the same, this could be a sign that the clutch or flywheel are defective, it could simply be that the clutch needs bedding in a bit (unlikely), so just do a few fast launches to see if that helps, or burnouts, whatever you like that is clutch-heavy. Sometimes clutches just thump and grumble a bit until they settle in. These are not torque-heavy engines, so you need to bumps the revs up to pull away, that’s how 5’s have always been. Try some fast pulls and whatnot, since the other possibility is that your clutch/flywheel are contaminated, although the fact that you said you end up pulling away fast sometimes with no shudder says to me that things are fine.
My instinct says either your gearbox is worse for wear, or that you might need to adjust your driving style. I imagine 5 speed gearboxes for the NC are pretty cheap and widely available and learning is free, just takes time.