TKD and karate stylists both form very large and diverse communities making it hard to meaningfully generalise about their technique, so I'll initially focus very specifically on the WTF-style dollyo chagi and the Shotokan mawashi-geri as demonstrated by Kagawa-sensei as in the videos the question links to.
The WTF kick in the video brings the kicking knee directly forwards past the supporting knee with the lower leg hanging down, then roughly keeps a line from kicking hip through the upper leg and knee towards the target while pivoting, then swinging the lower leg around to make contact. If you watch carefully the transition from 0:09 to 0:10, you can see the knee moves very little, and the rotation of the body is predominantly involved in reaching the knee forwards an extra foot or so rather than driving the knee strongly sideways and promoting power into the target. Consequences of this technique are:
power generation is pretty pathetic - most of the power comes from the thigh muscle; there's a degree of "slap" with relatively little follow through, but - unlike most of my comments here, that's in absolute terms and not relative to Kagawa-sensei's mawashi geri, which doesn't have a lot of follow through either
time between the preparatory position at 0:09 - where the defender still needs to be ready to defend against a front kick or side kick or even hooking kick, axe kick, or slapping/vertical kick, and it's still easy to step down and use a hand technique instead - is minimised, but
the shared forwards knee-raise movement itself is relatively easily jammed with feet or hands - especially if it becomes predictable
a back leg kick also needs more overall rotation of the hips than Kagawa-sensei's mawashi-geri, with much earlier and more obvious telegraphing of the intent to kick
it's impractical to deliver a really strong low-section kick or sweep using this preparation
recovery time is a mixed bag - because the leg isn't swung sideways strongly into the target, there's less to stop - especially if it doesn't meet expected resistance - and it can be drawn straight back towards the hip relatively quickly and easily, but the forward movement and leaning back make it harder to mix in punches, hand strikes, elbows etc, blocks, grabbing/trapping moves etc. while doing so
The mawashi-geri video is unusual in that Kagawa-sensei's approach to the technique is not practical for the average martial artist, requiring way too much flexibility, so in a sense you're comparing recommended-for-average-TKD-practitioner technique to barely-attainable-for-advanced-fulltime-Shotokan-practitioner technique. The mawashi-geri illustrated is distinctive for raising the knee sideways towards the shoulder without rotating the shoulders much. It's specialised for use at close quarters where it can come up and knock someone out without them even noticing a shift in balance or upper-body position that signals a kick coming. That technique is not intended to generate that much power - the idea is to strike the head and get a clean unexpected strike in past the guard.
This contrasts markedly with the much stronger kyokushin karate mawashi geri, or the Muay Thai kick, where the knee describes an arcs with more sideways movement during the kick. Indeed, their movement for a mawashi geri with the knee itself is more consistent with that used when striking with the foot, and their kicks can be targetted at any height low in the kick - they're very powerful at low section, unlike the WTF dollyo chagi. Clearly the WTF kick can't be changed into a sideways-striking knee kick if the opponent gets too close - it only moves forwards.