Ehm, beg to differ. Both healers in my static play on PS4 with controller. How they do it? I don't know, but they have no issue double weaving.
Even more so, a drg in our guild who plays way better than us (cleared dsr pretty quickly and last savage tier in week 1) - plays on controller.
So what i've heared it is pretty good, and it seems to work. No idea how as i will always prefer keyboard.
And true, they miss logs, but that changed when they started raiding with us, as we log and they can use analysis themselves. So it's not a big deal when it comes to raiding.
- - - Updated - - -
Triple CAN work, but if it fails you clip the GCD. As you said, all classes and rotations are designed with double weave in mind so there is no advantage to a successful triple weave (though i think there was some shennanigans with blm in shb) and you loose if it fails - so most guides will discourage them.