The sentence is awkward.
Drop “even the doctors,” and you get
No one thinks
which is obviously correct and idiomatic. “No one” is the subject and is singular.
When we add in “even the doctors,” we introduce a plural aspect that may suggest a plural verb. This is a grammatical mistake even though such mixtures of singular and plural fool even native speakers.
This mixing of singular and plural ideas is what makes the sentence awkward. Does it really mean much if the man who mows the lawn thinks you are healthy or ill. Surely what is important is what trained doctors think. What expresses the essence of the thought is
Not one doctor who examined you thinks you are ill.
There is no temptation to use a plural verb, and the thought is not encumbered with the irrelevancy of whether the mailman agrees or disagrees with those doctors.
This is another example that careless thought, or more usually hasty speech, induces grammatical error.