It sounds like primarily a level of service problem.
This might [0] be a case where having your own POTS [1] hardware makes sense. There is a vast expert technical infrastructure to support POTS and it is dedicated to reliability. The marketplace is highly competitive across market segments, by which I mean that there is heavy competition for two line systems and five hundred line systems. Above a certain threshold many systems are modular and provide 2x-4x linear growth.
Sure, maybe it won't scale out to a vast distributed team and you'll have to swap it out in two years and that will be painful. In two years your company will be swapping out phone systems if it is successful anyway...or it will be putting up with a system that's designed for the wrong larger scale until your company grows into it. That friction will probably reduce the the odds of getting there.
Telephony is hard enough to have given us Unix, C, Erlang, and information theory. For a business (and it looks like your company may be one) that will live and die by it's telephone system, controlling the hardware is consistent with Spolsky's approach to StackOverflow: it has its own servers because it can't afford to have critical infrastructure maintenance and repair happen on someone else's timeline. [2]
Good luck.
[0]: Or might not.
[1]: I'm cheating a little with "POTS", probably there's a VOIP component, but the big idea is a box in the closet with some telephony company's logo on it and a box of user manuals on a shelf.
[2]: edit - I might double down on controlling telephony as critical because of HIPPA.
[3]: edit - Outsourcing may make sense for non-patient call streams...e.g. sales and vendors and other generic business operations where a lower level of service might be acceptable.