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.