Low mass stars such as red dwarfs/brown dwarfs/ultra-cool dwarfs (wiki lists TRAPPIST-1 as a UCD - the differences are a bit arbitrary anyway) all tend to produce strong flares, so yes, the radiation stripping the atmosphere argument holds. Some stars are more active than others though, so you never know. Only way to know for sure is to wait for JWST to have a look in ~2019 or one of the 30-meter class telescopes with adaptive optics due online in the early 2020s.
As I understand it the issue is not that we don't think there are Earth-like planets around Sun-like (i.e. more hospitable) stars. It's just harder to detect planets that are so much smaller than their host star. As such we're currently detecting rocky planets around small stars and Jupiter sized planets around larger stars. New telescopes should improve this as well as allowing a better analysis of the possible atmospheres of planets.
* Pedantic point: White dwarfs are something different - rather than starting out as cool, low mass stars they begin as medium mass Sun-like stars. Once the hydrogen has been fused to helium they expand to red giants and then expel their outer layers to leave the core of the star as a white dwarf. Not a good place to look for planets as the red giant phase would have enveloped or at least fried them! We still have ~5 billion years left before the Sun reaches that point thankfully.