A researcher should offer a sufficient compensation package to get enough volunteers after explaining the risks. They should get independent medical advice too.

They can still randomize within the volunteers with treatment / placebo, and maybe use quotas, but they'd just have to extend their trial period until they'd achieved a measurably representative treatment and control group and enough volunteers to test the hypothesis to the required level.

This type of non-random sampling may very well have to be done anyway, for example if they needed the power to test efficacy and safety in all the potential dug interactions or co-morbidity scenarios. Not to mention any diagnosis requirement will also screen the sample which could be influenced by health care system resources and policies, not necessarily pure morbidity. So I think they can deal with non-random sampling in med research perfectly well.