I'm struggling to understand the restrictive/non-restrictive distinction for relative clauses modifying indefinite noun phrases.
The distinction seems very clear for definite noun phrases. It's a question of whether the relative clause is necessary to identify the noun phrase from the discourse/context or not.
A: The president…
B: Aah, the president.
A: The president, who had just toured the country, returned to Washington.
A: The man…
B: What man?
A: The man who won the election…
B: Aah, that man.
A: The man who won the election was instated in the white house.
For non-definite noun phrases, it seems to me that the same distinction doesn't apply, at least not in the same way.
I can think of two kinds of indefinite noun phrases
I'm looking for a TV that fits into my bathroom.
This use of "a TV" is similar to "any TV" or "TVs". In this case, the relative clause seems restrictive in the literal sense of the word. I'm looking for a TV, any TV, but with the restriction that it has to fit into my bathroom.
I cannot think of a non-restrictive relative clause that can modify this kind of noun phrase, at least in English. I think one would usually rephrase.
*I'm looking for a TV, which is entertaining. (Ungrammatical with a non-restrictive reading)
I'm looking for a TV, because TVs are entertaining.
I see that there are still cases where a restriction makes sense, as for non-specific noun phrases, e.g.
I saw a TV that fit into my bathroom.
I.e. I might have seen 100 TVs that day, but this was the only one that fit into my bathroom.
But there are also cases where this kind of restriction doesn't apply, e.g. in descriptions of scenery:
On the beach there were chairs that people could sit on.
This is the case that really bothers me. It seems that all the relative clause can do is add new information about the specific noun phrase (it's new to the context/discourse, so there is no identification involved), still I see people discussing them as if there is a meaningful restrictive/non-restrictive distinction. They tend to explain it with "How integral is the relative clause" and such, but that does not seem like a useful discrete test to me.
So finally my question:
Does the restrictive/non-restrictive distinction make sense for relative clauses modifying all specific indefinite noun phrases? If so, please explain it and provide syntactic or semantic tests that can distinguish, not vague tests like "is the relative clause integral to the sentence?".