Let's make the situation a little different. Could MSF pay a private surgery with investors to perform reconstruction for someone?
Could they pay the surgery to perform some amount of work they deem aligns with their charter?
Could they invest in the surgery under the condition that they have some control over the practices there? (Edit - e.g. perform Y surgeries, only perform from a set of reconstructive ones, patients need to be approved as in need by a board, etc)
Raising private investment allows a non profit to shift cost and risk to other entities.
The problem really only comes when the structure doesn't align with the intended goals - which is something distinct to the structure, just something non profits can do.
"Raising private investment allows a non profit to shift cost and risk to other entities."
for a suggestion of that.