This is a rather broad question, so I'd like to limit this to verbs, at least in this explication of the question.
Verbs take many forms and roles in sentences. Present participles can take the role of subject, object, adjective, and non-finite verbs in reduced clauses. Past participles can function as adjectives, passive verbs, and are used in the perfect tenses. The role of subject and object can be played by several types of words and word phrases (and even clauses); is it really necessary to reclassify them based on the roles they take?
Isn't classification independent of the role a word plays in a sentence; aren't all of the uses still properly classified as VERB in terms of their PoS?
I get so many questions from students about PoS, and I just want to tell them that these are all verbs, that role/usage/ and function do not necessarily correlate to parts of speech.
But in some cases they could. For example the words swim and look can be a noun or a verb.
I had a nice swim this morning.
She gave me a dirty look.
As a teacher, I would classify swim and look as nouns in these sentences.
But in other uses where the words take a distinctly verbal form in a subject or object I would still classify them as verbs.
Swimming is good exercise.
I like swimming.
I like to swim.
To swim to Kathmandu is a fool's errand.
I need to look for my dog.
I enjoy looking at old pictures.
Swimming, to swim, to look and looking are verb forms (PoS) that take the role of subject in or object in these sentences. At least for instructional purposes, they should not be classified as nouns just because they take the role of a thing in a sentence - they are still participle or infinitive forms of a VERB.
I understand that in linguistics this reclassification is called conversion or zero-derivation, but for students, requiring them to understand this concept seems onerous (as with many concepts in linguistics that are used in instructional or prescriptive grammars - too often in my opinion). I also get that even the noun swim is derivative (but one would need to understand etymology to know that, and this is well beyond the requirements of language learners.)
The human brain is a very capable language processor - we learn our first language without any comprehension of grammar concepts, and we can rely on this capability to learn second languages too, albeit with a bit of grammatical help but in a significantly reduced way than that employed for the purposes of linguistics. For learners, the primary thing they need to grasp is the nature of relations, which are primarily descriptive and adverbial, and the many ways they are produced and the different forms these relations take.
So, would I be committing some kind of linguistic or grammatical sin if I instruct my students to see these forms as their root PoS, and that these PoS can take multiple roles in sentences?