Actually, doctors in private practice are always required to provide effective communication under the ADA. So if an interpreter is needed for effective communication, they would have to provide terps absent a showing of undue burden. And even if they are able to demonstrate that terps are an undue burden, they are not excused from providing access - they still have the obligation to provide comm. access that is effective and does not result in an undue burden.
You are correct in that hospitals are places of public accommodation and would be required to provide effective communication. As to whether hospitals are recipients of federal financial assistance, that goes to the ability to bring a Section 504 Rehab Act claim, not the ADA.
Finally, while note writing may be effective for some deaf/hh people, it may not be effective at all for others especially when there'll be discussion over treatment options or whathaveyou for a serious DX. But certainly, if total privacy is important to the deaf/hh person, then they don't have to request a terp and go by note writing alone - although I would imagine they would have a hard time claiming ineffective communication later if they voluntarily waive their requests for an interpreter.