How does the loss of the conscience clause threaten physician autonomy? How would that affect me?
The loss of the conscience clause is simply one more nail in the coffin, but a crucial one. It marks the transfer of decision making from the doctor and patient to 'someone'.
The someone will be a government medicrat.
If the medicrat has the power to determine what services the doctor must provide, he/she has the power to determine what the doctor cannot provide.
If the doctor is forbidden to provide a service, the patient cannot receive it.
Where does this already happen? Look to Canada, where all medical care comes through the government system, and physicians are prohibited from operating outside the rules. Most Americans know that Canadians can wait months for many tests and surgeries because Canadian doctors have been prohibited from providing care except by the medicrat rules (although a recent court decision has led to cracks in the concrete). If someone wants the hip surgery sooner, they must leave the country rather than wait.
Currently in the USA, a government program or insurance company can decide to "not cover", that is, not pay for a service. The doctor may recommend, and the patient may decide to have a service and pay for it out of pocket. Although this can be inconvenient and expensive, the final decision is made by the patient. The patient preserves his/her autonomy, as does the doctor.
No national health plan can pay for everything, all the time, for everyone, although the politicians will pretend for a time. As expenses run away, the pressure to lower costs will lead to intensified hidden rules and restrictions (covert rationing) and eventually to explicit rationing. The current "not covered" will increasingly morph into "not medically necessary". Will it become forbidden?
Insist that any plan protect this freedom: that when your surgery/test/procedure/consultation is denied or delayed, you have the liberty to pay out of pocket without leaving the country.
Otherwise, expect your bladder difficulties to be treated by Vogon poetry.