The definition of 'clause' in modern grammar: construction vs function

Most modern grammars recognize verbless clauses such as the boy on the roof in (1) and on the roof in (2):

(1) With the boy on the roof, they feared he might jump off it.

(2) When on the roof, he might jump off it.

In these modern grammars, thus, having a verb is not a necessary condition for being a clause.

Having a verb is not a sufficient condition for qualifying as a clause, either. In (3), for example, frequently asked is a verb phrase (VP) containing a verb ‘asked’ (in the form of a past participle) but is not considered a clause in the same modern grammars.

(3) Each week, we answer frequently asked questions about life during the coronavirus crisis.

Therefore, I conclude that having a verb is neither a necessary nor a sufficient condition for being a clause in those modern grammars. With this in mind, let’s look at this pair:

(4) The boy who was on the roof jumped off it.

(5) The boy on the roof jumped off it.

They mean the same thing but, in those modern grammars that recognize verbless clauses, who was on the roof in (4) qualifies as a clause (a relative clause) whereas on the roof in (5) does not.

Note that in (1) and (2) above, the boy on the roof and on the roof are classified as clauses not because of their internal construction but because of their external functionality. And also note that from (1) through (3), having a verb has no logical connection with being a clause. Then, why is it that who was on the roof is treated as a clause in (4) whereas on the roof is not in (5) when they seem to have the same function of modifying the noun boy?

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By "modern grammars" I mean descriptive grammars compiling the past sixty years or so of linguistic research. "Modern grammars" are distinguishable from "traditional grammars", which are prescriptively established without the help of extensive linguistic research.

Examples of modern grammars include:

1. A Comprehensive Grammar of the English Language (by Randolph Quirk, Sidney Greenbaum, Geoffrey Leech, and Jan Svartvik)

2. Longman Grammar of Spoken and Written English (by Douglas Biber, Stig Johansson, Geoffrey Leech)

3. The Cambridge Grammar of the English Language (by Rodney Huddleston, Geoffrey K. Pullum)

4. Oxford Modern English Grammar (by Bas Aarts)