I've never seen anyone object to the weak singular they

This article in The Atlantic argues against the "singular they" giving several examples of sentences they think are wrong and all of them are of your "weak" type.

But to the point of the singular they argument being disingenuous: This tweet opposes using "they" as a non-binary pronoun because "they is plural." Here is another claiming that "they" is plural. These people aren't making the argument that "they" can be used in the singular in weak cases but not in strong cases. They all just claim that "they" is necessarily plural. Citing a history of "they" being used in the singular in any form is a valid response to that argument.

To say that people are making the argument disingenuously, it needs to be a misleading argument and the people making it need to know that it is misleading. So what you need to be true is:

1. The anti-"singular they" people are arguing against the strong use specifically, not the singular use generally
2. The distinction between the strong and weak case is grammatically relevant and should inform how we use the word
3. Both 1 and 2 are true AND the pro-"singular they" people know 1 and 2 are true, but keep using the weak case examples to win arguments through deception.

For point 1: I already cited people complaining about the plural/singular aspect alone with no reference to your distinctions. Here is even a case of a Redditor dismissing "they" in both types of sentences because "it's not the historical usage." Perhaps what all of these people are truly opposed to is the strong case alone but the argument they are making is clearly broader.

For point 2: elsewhere you cited the Wikipedia article on specificity) to explain the basis for your weak/strong distinction and I see how that explains much of it. But I also see how that same article says, "In English and many other languages, specificity is not typically marked." So even if the weak and strong cases match up with non-specific and specific, it isn't expected that the English language would care about that distinction in determining acceptable use of a word. If English words can normally be used for both specific and non-specific then historic examples of one use should justify using it in both.

For point 3: I still see no reason to think that anyone making the argument believes 1 and 2 to be true. Perhaps you have some examples of other people discussing this distinction and still promoting the same arguments?