I never thought about this! Yes, it's certainly confusing. Sometimes we use the definitive article to refer to a class, or group, or category. The category itself is singular because it is only one category, and I guess use the definite article because only one of it exists in the world. So, we might say "The killer whale is the best studied of all whales. Here, "the killer whale" does not mean one particular individual whale. It does not mean "Namu" or "Shamu" or "Corky II." It does not mean "that whale in that aquarium over there." It means "the species Orcinus orca." It means "Scientists have published more papers about whales of the species _Orcinus orca_ then they have published about whales of the species _Physeter macrocepahlus_." Similarly, in the first saying, "the early bird" means "the category of all birds that are early." In the second, it means "the category of all doctors." We can rewrite your sayings so that they make sense, mean the same thing, and don't use the definite article--but the sayings are what they are. The rewritten versions are: "Early birds catch more worms." "An apple a day keeps doctors away."

In the first case, I think "the" emphasizes that after the early bird catches it, there is no more work for the lazy bird. If you'd say "a worm", it's less clear that the late bird is not catching worms. In the Netherlands, people typically go to a fixed doctor unless they move to another city (this doctor will refer them to specialists as needed). Perhaps it's the same in England (or was when the proverb was first used)? In that case it would make sense to refer to "the doctor", since you're not going to just any doctor. I'm not sure, but I don't think there's a strict grammatical reason; "a worm" and "doctors" aren't wrong, that's just not how the proverbs go.