I think a non-restrictive relative clause cannot be used when the word before the clause is indefinite - am I wrong?

Question

Relative clauses are categorized into two types in terms of restricting meaning: restrictive clauses and non-restrictive clauses.

In my view, a non-restrictive relative clause cannot be used when the word before the clause is indefinite (an, an, plural forms without "the" or "those" or "these", etc).

I think this way because an indefinite word has to be narrowed down in some way in meaning.

If not, the attribute that the clause gives is applied to all the samples.

For instance, "a lion is big and eats meat" actually means "all lions are big and eat meat."

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However, I found this case here, and it contradicts my idea, and I can't explain why.

Example 1:

My doctor is trying a new testing method, which had a 70% success rate.

Doesn't Example 1 basically mean that all new testing methods, no matter which one, had a 70% success rate?

(Source of Example 1: https://grammar.yourdictionary.com/parts-of-speech/pronouns/relative-pronoun.html)

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Let's look at other examples I made here.

Example 2:

My brother is looking for a room to rent, which is cheap enough for a student.

Doesn't Example 2 mean that a room to rent is always cheap enough for a student?

Example 3:

My brother is looking for a room to rent which is cheap enough for a student.

Example 3 is what I think works the best because it means that there are different types of rooms but he is looking for a specific type of room to rent that is cheap enough for a student.

Answer 1

Your hypothesis is incorrect. The referent of a relative clause can be unique (and thus take a nonrestrictive modifier) even if it begins with an indefinite article. That is a possible interpretation of your example #1: The doctor is presumably only trying one new testing method, so it does not need to be restricted further. (By the way, it is also possible to write "My doctor is trying a new testing method that had a 70% success rate." However, the meaning would be different; that would probably mean that the doctor has already tried other new testing methods, which presumably had different success rates.)

There are many other examples, e.g.:

I'm sure that you have a belly button, which is a reminder of your birth.

There is no need to restrict "belly button", because we know exactly which one we are talking about despite the indefinite article. This sentence would be strange if the relative clause were restrictive; that would imply that you might have multiple belly buttons!

Your suggestion that a nonrestrictive relative clause cannot be used after "plural forms without 'the' or 'those' or 'these', etc" is also incorrect. E.g.:

I like sports, which my sister hates.

"Sports" means all sports, and there is no need to restrict it any further. The relative clause could be restrictive, but then, of course, the meaning would be different.

You are correct about examples #2 and #3; because he is presumably looking for a particular kind of room, the relative clause works better if it is restrictive.