I'm not convinced that your expression "all the strawberries he does have" is a recognized way of disambiguating the expression that you say is ambiguous: "all the strawberries he has." When would we use the expression "all the strawberries he does have"? As far as I can see, only in special contexts such as this one: "He doesn't have all the strawberries in the county. But all the strawberries he does have are organic." In that example, "does" isn't used to signal the indicative mood; instead it's used merely to emphasize a contrast.
Nor am I convinced that "does" + infinitive always carries existential import (i.e., implies the existence of at least one thing satisfying the verb phrase). Consider:
(P) "All the intelligent extraterrestrials our galaxy does contain are extraterrestrials."
Again, P will sound awkward except in a context such as this:
(Q) "Our galaxy may not contain any intelligent extraterrestrials. But all the intelligent extraterrestrials our galaxy does contain are extraterrestrials."
Whether or not you believe our galaxy contains intelligent extraterrestrials, it would be wrong to deny the second sentence in Q, wouldn't it?