It's no problem leaving HSS as default.Your flash works much harder during HSS, and produces less light while doing so.
So there are at least two problems: 1) the increased power consumption will wear down your batteries faster, and 2) the increased power consumption will heat up your flashes faster.
It seemed the core of his question was what happens at 1/250 s (Xsync) and slower, and in that case the flash fires normally even with HSS set. The flash only fires in HSS if a shutter speed higher than Xsync is selected. I wouldn't consider increased power consumption a 'problem' although it's good to be aware of it (one should always carry extra batteries if relying on flash). I don't know that overheating would be an issue, but the 600EX-RT has an overheat warning (the display backlight turns red).
But it might be worth knowing why the OP wants to use HSS, there may be other solutions. The flash is much faster than 1/250 s, and can be used to stop action. Alternatively, if using fill flash with a very wide aperture in daylight, an ND filter can knock down ambient light to Xsync speeds.
I would like the possibility to stop action with HSS, but also to overpower daylight. I have never used ND's for this, but will certainly give it a go. None of my lenses are F1.2 (where I have read about this technique before), but three of them are at 1.4, so I guess the same goes for them.
I don't think you're going to stop motion or overpower daylight in HSS.
A normal flash duration is very fast, and can "stop motion". But HSS rapidly blinks for the entire duration of of the exposure, so it's not really any different than ambient. The flash won't stop motion any more than the normal exposure would (when above max sync).
HSS also cuts into your power, considerably. Flashes have enough trouble overpowering sun at full power; in HSS you're not going to overpower very bright sunlight. Maybe bare, close to the subject, but you will notice the power reduction.