Both FUT and FUE have advantages and disadvantages.
One big misconception with FUE is that you can easily shave your head in the future. This isn't true for many people especially if you've had a large number of grafts extracted ( even without overharvesting) For many people all those small dot scars are visible even if a 0.9mm or 0.8mm punch is used (I've seen many) Many factors affect the visibility of the scarring including skin tone etc. Even if you don't scar there is a space with no hair which is nearly as noticeable.
I'm talking about shaving your head here, but if you trim with clippers and leave your hair at least 2-3 mm then you can get away with it. In any case you can always wear your hair shorter with FUE than with FUT. Also you can camouflage FUE scars with SMP much easier than with an FUT scar if you want the total shaved look.
FUT is a good option if you need a very large number of grafts. Also FUT surgeries only take one day, if you need large numbers of grafts with FUE you'll need a two day surgery or even multiple surgeries spread over months/years to achieve what can be done in one day with FUT
FUT is more aggressive and painful but cheaper, FUE is less painful but slower to perform and for larger graft numbers will require a two day surgery.
Both procedures require you to shave the recipient area for a good result. The clinics that tell you you don't need to shave just don't want to put you off and lose your money so they tell you what you want to hear.
Whether the doctor or a tech does the implanting doesn't depend on the technique it depends on the clinic. Most clinics nowadays use techs to do all the implanting as they can be rotated to avoid fatigue. It's really difficult for the doctor to do all the work unless it's a very small surgery so I would be wary of any clinic using that as a selling point.
The only exception to this is if they use implanter pens which I believe are quite common in Korea. In this case its the doctor who should do the implanting as it requires making an incision with the pen which in most places only a doctor can legally do.
In your case you say you still have good coverage on top so my advice is DON'T do any transplant in that area, you could do more damage than good and end up with no net benefit. You should try meds first.
In summary FUE is safer/less invasive and better for most people unless you need really large graft numbers. At a good clinic the yield should be very similar. And if the transplant fails or the result isn't good you'll regret it much less with FUE than if you have a long scar at the back of your head for nothing.
