Is it true that the tones of some of the most common words are glossed over in regular speech/turn into neutral tone?
I discussed quasi-neutral tones in the fifth point of my answer to another question here. Salient points include:

Usually, a stress is placed elsewhere in the sentence. Such words (edit: usually frequent words / function words, i.e. 虛詞) are therefore weakened in their presence. See the three examples in my orginal answer for more details.
The tone sandhi is facultative (i.e. the citation tone and quasi-neutral tone coexist), not obligate.


They are not glided over when, e.g.:

There is pragmatical reason to place stress on them. You emphasise 是 in 我是學生 when e.g., someone is doubting your identity as a student. Interestingly, 學生 may become relatively toneless in the presence of a stressed 是, especially in rapid speech.

There is contextual reason to place stress on them, e.g., when you are delivering a speech (especially the 朗誦 kind that values intonation, more so than 演講). Otherwise, in oral exams, if you are expected to be conversational, gliding over is permissible, even more natural.