Every martial art has its own philosophy which influences what is emphasized. That philosophy is in turn influenced by:
- Terrain of the region it grew up in
- Available weapons (including strikes as weapons)
- The martial arts of the surrounding areas (i.e. common enemies)
If the terrain is a loose gravel, or many unpredictable sink holes in the ground, sliding feet will help discover those areas and compensate more quickly. There's nothing more useless than falling down or twisting your ankle in the middle of a fight. Areas with more predictable or stable terrain will enable more picking up your feet.
Rapier is designed for thrusting strikes. The point is tempered to a harder steel than the body. The thrust being the kill technique. In order for that type of weapon to work, you need speed. Rapier arts are designed around that philosophy, so the combination of picking your feet up, and the types of attacks and defenses available are influenced by the sword.
Iaido and kendo are built around a katana which is designed to slice on the edge. One side of the blade is tempered to a harder steel, providing a sharp cutting edge while the main body of the sword is tempered to a softer steel providing strength. While there is a thrust technique, it is only one of the 8 strikes and the least preferred. The katana based sword arts use the hip as part of the cut.
Even in a broader sense, Japanese martial arts tend to favor a philosophy of "no wasted movement". If I can close the distance by barely moving my feet just as quickly as someone who picks their feet up, then I would consider picking the foot up to be wasted movement. I can use that energy for the hissatsu (final/lethal strike). The sword arts were also influenced by the open hand arts (and vice versa), so you'll find many similarities.
What all this means
Every martial art has a philosophy, and not all philosophies are compatible. Training for pure speed is very different than training for power. Some arts favor speed because a strike, while it may not disable the opponent, will distract or disorient them allowing for more strikes. Some forms of kung fu are based on that the idea that you overwhelm the opponent by a barrage of strikes. Others are based on the idea of "one hit, one kill", which is a philosophy in Tae Kwan Do.
The philosophies behind the rapier fighting would be at odds with hapkido. You use different tools. You have very different cultures and environments where the martial arts grew up. Eventually, you will gravitate toward one of the more basic philosophies and embrace it. That doesn't mean that one is inherently better than the other. It means that one is more useful in certain circumstances than another.
The basic law of the jungle is: whoever is most prepared for a fight will win. Some of the factors that can favor one approach vs. another can include the terrain, whether you are indoors or out, elevation, etc. However, all things being equal, the one who has trained the best for that fight will win 9 times out of 10. (the remaining 1 time is for someone who just gets lucky).