I have seen quite a few sentences that look like this in 18th and 19th century literature:
There was hardly a lawyer of repute but took up the question, and had an opinion as to Lizzie’s right to the necklace.
― Anthony Trollope, The Eustace Diamonds (1871)
And as for canes, hardly a man but has a half-dozen or more of all styles, colors, and weights.
― Mark Sibley Severance, Hammersmith: His Harvard Days (1878)
And after this matter it is said we come to the abstract idea of man, or, if you please, humanity, or human nature; wherein it is true there is included colour, because there is no man but has some colour, but then it can be neither white, nor black, nor any particular colour, because there is no one particular colour wherein all men partake.
― George Berkeley, The Principles of Human Knowledge (1710)
These buts are very akin to that doesn’t or doesn’t have. (Note that in the first sentence, Trollope means that reputable lawyers ᴅɪᴅ take up the question, as revealed by context, since he goes on to list a couple of solicitors and attorneys with opinions.)
Then there are a few easier explained that use but strictly instead of that:
I thank your Lordship for your invitation to Chilbolton, but I fear it is impracticable this year, It is not impossible but that next year I may have the honour of waiting on your Lordship at Sᵗ. Asaph, If I go to Ireland I certainly will go that way.
― Sir Joshua Reynolds, Letter LXXVII¹ (1784)
But using that rule on Trollope’s sentence would turn the sentence on its head completely.
I would like to learn more about the former usage, and what kind of rules explain it.