Before this question is marked as a duplicate, understand that I am not questioning the classification of a non-finite clause nor asking the generic reason for said classification. I know that the subject is implied; however, I don't understand why that is sufficient justification.
Let's look at an example that doesn't contain a non-finite clause:
He watched from the rooftop.
In this sentence, we have a finite clause ('He watched') and a prepositional phrase ('from the rooftop'). The preposition phrase is functioning adverbially, explaining the way in which he watched. Now, I would like to know how this is any different from the next example:
He watched to find his target.
If I have interpreted the definition of a non-finite clause correctly, I believe that 'to find his target' (something I would ordinarily class as an adverbial phrase) is one of them. According to what I have read, the subject 'he' is implied by the matrix clause. However — and I know this is wrong (I just want to understand) — is it not implied that 'he' is the one watching 'from the rooftop' in the previous example? Why is the first example a phrase, while the second is a clause?
I would also like to add that most accessible English learning resources (available online) scarcely acknowledge the existence of a 'non-finite clause', which makes this especially confusing.