This question is one of the oldest debates in this hobby. As is the case with any debate, there is no correct answer. I have always flown with thumbs only, but I started using thumbs/finger with my quad. As others have said, you generally get more precise control with the thumb/forefinger method. Most multi rotor and heli pilots I know use two fingers on each stick. This tends to work well, since these pilot are moving both stick in all directions for most of the flight. I think the thumb/forefinger method is also popular with 3D pilots. Like multi rotor and heli pilots, they are manipulating both sticks continuously, for most of the flight. Many airplane pilots who simply "bank and yank" will use to left stick very little after takeoff and before landing.

My main complaint with the "thumbs" method is the fact that the ends of the sticks start to make my thumbs sore after a while. This is particularly noticeable when I am practicing on my simulator for long stretches of time, as I never fly for more than 15 minutes at a time at the field. I suppose this problem could be mitigated by adding something soft to the tips of the sticks. That said, the rough ends that are found on most sticks help to avoid the thumb from slipping off the end of the stick. Furthermore, it can be somewhat difficult for some people to fly with their thumbs if they are wearing gloves, if the inside of the glove tends to slide against the skin. One pilot I know has a pair of gloves with cut-outs on both thumbs, to allow direct contact with the sticks.

A neck strap or tray is a very good idea if you use thumb/forefingers. The tiny transmitter that came with my quad is small and light enough to be supported by my pinky/ring/middle fingers, but I would always use a neck strap with anything that is heavier or larger. I personally use a strap at all times with my Taranis, even when I am using my thumbs only, just for the added safety margin.

I agree. Most people have more dexterity in their thumbs and forefingers, as compared to the other fingers. Using the pinky/ring/middle fingers for the switches and knobs is doable, but usually more awkward than using the pointer fingers for these functions. It is harder to reach the multitude of switches and knobs on my Taranis if I am "pinching" the sticks. Even the sliders on the sides of the transmitter case are more difficult to use.

In the end, every pilot chooses the method he/she prefers. No one is "right" or "wrong" for the choice they make. I have never heard of an accident that could have been prevented by using one method over the other.