One criticism I could think, if the experimentation has side-effects, the consequences could be catastrophic. Might be best to save that for time out of office.

This logic doesn't hold up. Should members of congress also get preferential access to experimental treatments? What about governors? We should not give any preferential access based on the status or public office of the patient.

If the president, as a citizen, requests an experimental treatment for compassionate use, that should be considered the same as anyone else.

The difference here is that the president would clearly be abusing his office to get the treatment, and this should be disclosed in his medical evaluations.