But words on their own are not any part of speech. The word 'up' is, in itself, not any part of speech. It can be used as a preposition, verb, noun, adjective, etc. Similarly, the word 'shut' is not, in itself, a verb.
But so much of this discussion and apparent confusion comes from mixing terms from morphology and syntax. These are different subjects with different lexica.
How a word is operating within a sentence is a matter of syntax. There the words "verb" and "adjective" can be confusing. It would be better to speak of "simple predicates" (or "gerunds" or "participial phrases" or "infinitive phrases") and "modifiers".
Morphology has little to do with it. It's usage that counts. In my Webster's Third (1961), the word 'disembowel' is listed only as a verb, because it has been used only as verb so far in recorded English. 'shut' is listed as a verb, noun, and adjective, 'up' as an adverb, adjective, verb, preposition and noun.
These latter terms will not be used by a dictionary, which limits itself to morphology, although it may give some contexts to show how a given word can be used as more than one part of speech.
In none of the first four grammars I have just taken from my shelf is 'to' used as part of the infinitive, Hudleston and Palmer (2002) use bold italicised font for the infinitive:
Regarding my naming the verb "to shut" as "to shut", in my experience that is what linguists, careful grammarians, and careful teachers of foreign languages do. They name verbs by their infinitives. If you simple refer to the verb "to shut" as "shut", it could be the present, past, or past participle that you are talking about (or the latter being used as a modifier -- adjective, if you will).