The teacher's use of "dead words" does not make sense. Often in English, so-called dead words are words that have been, metaphorically, used to death. The phrase "kind of" would be an example of a dead phrase. "It was really very kind of cool" makes no sense otherwise. "Kind of" was a softener, or a qualifier. It has not become its antonym (i.e., it has not become an intensifier; it has not fallen to the fate of a word like "literal"). It is just meaningless. It went from an overused softener (because no one wanted to sound arrogant or take the risk of being definitive) to just being a verbal tic. When we soften phrases, we add "sort of", yielding "it was kind of sort of fun" (softened) as opposed to "it was kind of fun" (the new neutral) or as opposed to "it was really kind of fun" (the new intense).
But should/could/would still have lexical content. You cannot take them out of sentences and expect the resulting sentences to carry the same meaning or even for them to make grammatical sense.
There is no reason to call a word like "should" a dead word. There is no replacement for it. It can be overused, but this is different from being a dead word. "You should leave now." can change to "Seriously, it is imperative that you vacate immediately!"
If we totally avoid words like should/could/would we end up with unwieldy "purple prose" and the excessive repetition of other constructions. Saying someone should not use "should" is akin to saying someone should not use "not" or "the" or "have" in the phrase "I have scrounged the depths of this lake for twenty years in vain, hoping to find the remains of the S.S. Toby; not a day goes by where I don't consider giving up."
Let's look at the following: "I really should have checked the seat when I got up; this was the third wallet this year that I lost because it fell out of my pocket in a taxi cab." Now consider the following avoidance of "should": "This being the third wallet this year that I lost from it falling out of my pocket in a taxi cab, my getting out of taxi cabs necessitates my checking the seat before leaving."
A better candidate for removal is "very". For example, "I was very upset" can become "I was devastated". Also, trying to avoid the verb "be" is a good tool, and the excessive use of passive voice is my bug bear. See the following sentence "The news would have devastated her on any other day, but nothing could bring her down today." I have no idea how this teacher would suggest changing the previous sentence, or even this one.
You often do not need to be skilled in your content area to be a schoolteacher. It would be nice if all were so trained, but many are only trained in pedagogy (not to diminish this!). Generally, just a general college education is enough. And even then, you do not need very many subject courses to get certified to teach a certain subject area, even at the high school level. (This assumes you are somewhere in the USA, where standards vary greatly from state to state.)