Are patents appropriate in taxpayer funded academic settings?

Yes it is ethical.

The patent will typically be owned by the institute and it will get most of the revenues (this is why they encourage it). The inventors usually get some form of compensation. Revenues from commercialization is partially how your salaries can be payed and could in fact save tax-payers money.

More importantly, developing a new diagnostic or therapy will in the end improve the lives of the general public (and even save lives in many cases). Currently, due to the immense costs involved, the only way to get a new therapy to the public is in a commercial setting.

I understand the answer above, but a follow on question I'd ask is: is it ethical to keep this 'good idea' away from other scientists, who indeed might be able to make better progress on it than only the idea generator themselves?

It seems to me that the end goal of all of this should be patient care, and to best benefit this, my opinion is that publication in an open access forum is the first, and most important way forward. I understand that without the protection of a patent, businesses can't be built, and indeed, that our whole society is based on this concept of capitalism. But, it doesn't jive with my sensibilities as a physician.