What follows is relevant most often to kicks in which your opponent is directly in front of you, or perhaps slightly to the right (if in orthodox stance), and when your opponent is within close or medium range.
Some of the best rear leg roundhouse kicks I've seen have been executed without pivoting (or very minimal pivoting) of the front foot.
The technique requires the striker to position their front foot at roughly 45-75 degrees to the outside, and the rear leg simply launches and retracts without much front foot movement.
This requires a certain degree of hip and knee mobility if proper weight transferral is to be achieved and if injury is to be prevented, but the advantages are considerable:
It requires less energy, as less movement is required,
It maintains greater defensive and offensive capacity throughout the motion, as your foot retains the optimal position from which to initiate a variety of follow-up movements.
It enables faster strikes and facilitates quicker repetition of the same strike. A kind of torque-spring is created in the body which weakens when exaggerated pivoting of the front foot occurs. If you compare both techniques when doing drills, you will feel this.
Buakaw illustrates these points beautifully with this burst of strikes (0:31). You will notice however that when he transitions (0:41) to right/left combinations and is chasing his partner with close to maximal power kicks, his foot pivots a lot more.
Again, we find an example of how neither technique is more correct than another. Both have application and should be incorporated into your training.
In relation to your point about it being 'not a good feeling' when you pivot, this likely occurs for a couple of reasons:
Carpet is a non-ideal high-friction surface. Until you condition the balls of your feet to that particular surface, you will likely experience blisters and discomfort. I have resorted to training on carpet in the past, and your feet do become conditioned to it if you train on it regularly. You simply need to endure a couple of weeks of discomfort and remain on the lookout for any infection whilst your feet are responding with blisters, which will almost certainly pop. Wash your feet after these sessions and apply antiseptic if necessary. You may also want to try foot wraps.
Training in shoes if you are hoping to practice Muay Thai in a ring is not ideal, simply because to do so is to diminish the specificity of your training. You are unable to grip the floor in shoes as well as you can in bare feet, and this results in you having to make allowances for the resulting lack of stability, which will often mean a loss of repeatable power, range of movement and retraction speed, although the extra weight on your feet may improve your kicking fitness slightly and reinforce aspects of balance which translate well to bare-foot techniques.
It is desirable to train in shoes if your goal is to be 'street' ready. If this is the case, you simply need to train frequently in shoes, so that you become accustomed to the instability you describe. Consider buying shoes which have a sole more suitable to your environment. If you can't afford to do so, you will simply have to adapt to what you have.
The more friction that exists between your foot and the ground, the more you will have to unweight yourself in order to pivot properly and without injury. This does not however require that you begin to "bounce", for the reasons described in an answer to your recent question about bouncing.
The more you are unweighted, the more vulnerable you are. You also typically want to be fully-weighted or close to fully-weighted at the moment you make impact with your opponent if maximum power transferral is your goal.
- Experiment with minimising foot rotation. This will minimise the problem of friction.
- Train as closely as possible to how you expect to fight (ie: barefoot vs shoes, ring mat/gym floor vs asphalt).
- If you must train on carpet, consider wrapping your feet or else be wary of infection.
- If you want to train in shoes, try to obtain some with soles that aren't too sticky, but which have useful grip.
- Learn to unweight yourself as you kick, to the minimum degree necessary in order to achieve unimpaired rotation, and to re-weight at the point of impact.
- Try not to bounce.