Is there any logic behind continuing to distinguish 'gerund' from 'present participle' in traditional grammar?

Question

To an earlier question "What's the difference between a gerund and a participle?", there is a consensus among the answers there, and I quote the most upvoted answer:

A gerund is a form of a verb used as a noun, whereas a participle is a form of verb used as an adjective (or as a verb in conjunction with an auxiliary verb).

The bracketed portion is not stated in the other answers, and here I'd like to focus on the claim that 'gerund', unlike 'participle', is used as a noun. Hence the bracket.

The most upvoted answer there cites a wiki on 'gerund', which in a relevant section says:

An -ing form is termed gerund when it behaves as a verb within a clause (so that it may be modified by an adverb or have an object); but the resulting clause as a whole (sometimes consisting of only one word, the gerund itself) functions as a noun within the larger sentence.

For example, consider the sentence "Eating this cake is easy." Here, the gerund is the verb eating, which takes an object this cake. The entire clause eating this cake is then used as a noun, which in this case serves as the subject of the larger sentence.

(Boldface mine.)

In the wiki, 'noun' should be replaced with 'noun phrase' because it's not a noun but an NP that "serves as the subject of the larger sentence".

With this correction in mind, here's my analysis of the wiki example:

(1) [Eating this cake] is easy. [Eating = noun??]

Here, it's not Eating itself but the entire verb phrase Eating this cake that is used as an NP.

(2) The man [eating this cake] is my brother. [eating = adjective??]

Similarly, in (2), it's not eating itself but the entire verb phrase eating this cake that is used as an adjective phrase (AdjP).

Note here that the verb phrase eating this cake can be said to be used not as an adjective but as an AdjP because an adjective normally cannot post-modify a noun but an AdjP can.

*The man hungry is my brother.

The man [hungry for cake] is my brother.

All in all, eating in (1) is not used as a noun, just as eating in (2) is not used as an adjective. If anything, eating in (1) and (2) is a verb used as nothing other than a verb.

This becomes even clearer when you compare it to similar infinitival constructions:

(3) It is easy [to eat this cake] . [(to) eat = noun??]

(4) The man has the ability [to eat the entire cake]. [(to) eat = adjective??]

Even in traditional grammar you don't distinguish '(to) eat' in (3) from '(to) eat' in (4) by assigning different terminology, which makes the practice of distinguishing 'gerund' from 'present participle' all the more illogical.

Therefore, it's a gross mistake to distinguish 'gerund' from 'present participle' simply because:

a gerund is a form of a verb used as a noun, whereas a (present) participle is a form of verb used as an adjective

QUESTION

Since this distinction doesn't work, is there any other logic behind continuing to distinguish 'gerund' from 'present participle' especially at a word level even in traditional grammar, if traditional grammar is to remain anything logical?

If there is no such logic, which I doubt there is, why does traditional grammar still use the 'gerund'/'present participle' dichotomy for the very same form 'V-ing'?

EDIT

For those who argue that the cited question/answer isn't a good enough source to represent 'traditional grammar', here's an excerpt from a grammar book called Common Core Grammar: High School Edition (2015) that says in part:

Students must not make the mistake of thinking that every "verb + ing" is a participle. A "verb + ing" is a participle only when used as an adjective; it is a Gerund when used as a noun: that is, as the subject of a verb, object of a verb, etc.

Apparently, the book is written by an English teacher in America for American high school students.

Also, The Practical English Usage by Swan says:

When -ing forms are used as verbs or adjectives, they are often called 'present participles'...When they are used more like nouns, they are often called 'gerunds'.

So there's no denying that many, if not all, grammars based on 'traditional grammar' or 'school grammar' make this illogical distinction between 'gerund' and 'present participle'.

Answer 1

While searching this site for a related topic, I've come across this earlier question: Should “gerund + objective” be modified by adjectives or adverbs?

There, @tchrist gave this great answer, which basically says it'd be better to ditch the term 'gerund' altogether, and I thought I might as well post an answer myself based on that answer, and I quote:

It may surprise you to learn that “gerund” isn’t so useful a term as you might think, and you don’t even need it. You’ll find that the analysis becomes far easier, both in this case and in more complex ones, if you discard the term entirely and stick strictly to parts of speech: verb, noun, adjective, adverb. If you want to discuss its broader syntactic role in the grammar as a constituent, then we use other terms for those constituents than parts of speech.

That said, if anyone can come up with a book or a paper that touches on this very issue, please go ahead and post an answer.