I don't know what other airlines do. Quite honestly, once the take off roll has started I'm surprised that anyone would entertain the concept of even answering the phone. A take off roll lasts for about 60 seconds. Most likely some of that (perhaps up to 20 seconds) is already past the refusal option. So someone in the cabin has to become concerned enough to actually ring, the phone has to be answered (find the right switch and select it...it's not in your field of view and won't be on). Conversation has to ensue. Then the gist has to be passed to the captain. And then he has to decide, based on third hand information. The time window is miniscule and the information too vague.
It's not a case of not wanting to know. Getting airborne is almost certainly the BEST option.
There are some scenarios where an abort at any speed might be best, and the one that comes to mind is what you might describe as 'self rotation'. If the aircraft attempts to rotate, by itself, well below Vr, then you might be looking at a CofG shift, or large loading error. That's something that you can't fix in flight.