Edit-preface: I am not a grammarian. I don’t know what the technical names for the different types of “to” are or if they are even recognized as distinct by experts in the field.

English is does indeed use “go” to mean “go do a thing”, but not with directional “to” (as in “go to the library”).

“Go run!”, “Go running”, “I’m going running”, and “I’m going to run” are all valid uses. (In that last case, the “to” is not a directional “to”, but is actually part of the infinitive verb “to run”, as in “I want to run”). However, you wouldn’t say “Go to run!” to tell someone to run.

“Go to run” could make sense with a causal “to” (“Go, in order that you might run”) but that separates “go” and “run” in to separate actions. Causal “to” is the “to” in “push to open” and “press F to pay respects” this is not the “to” in “go to sleep”

“Go to sleep” feels like it is in the directional sense, like “go to bed”

Edit: Now you’ve got me thinking. “Go to sleep” and “go to bed” are a little unusual . “Go to [location]“ without an article is usually reserved for proper nouns or pronouns (“Go to France”, “go to Curicó”, “go to Walmart”, “go to John“ “go to her”). When the location is a general noun, you usually use an article or a proper/pro-noun in the possessive form (“go to a restaurant”, “go to the party”, “go to Bob’s house”, “go to your room”). So what makes “bed” and “sleep” so special? The only other case I can think of at the moment is “go to ground” and that is different because it is an idiom, and the rule for idioms is “they mean what they mean”

Edit-edit: meals don’t use an article either: “to lunch”, “to dinner”, “to breakfast”.

Edit-edit-edit: AAAAAH! It applies to some other prepositions too: “in bed”, “at lunch”; but not “under the bed”. What is going on‽

Edit-edit-edit-edit: Causal “to” might be a use of the infinitive case?

Edit-edit-edit-edit-edit: “go to work” does not use an article either.