What overarching category do "street," "road," "avenue", etc. belong to?

There are categories, and categories of categories, and so on, and any of these categories may or may not have labels already in the language, as Colin pointed out. If a set of words has a word for that set, that word is called a hypernym.

And the semantic category depends on the particular collection you want to name (the collection may not be coherent).

But for these three words, I find that the best encompassing hypernym is

road or roadway,

even though 'road' is one of the things you want as a subcategory, it works as a generalization of them all, a large two-way ...thing... to travel on. (A word that is its own hypernym is an autohyponym or autohypernym). It doesn't have to be paved but a 'path' is too small to be included. I'm not sure about 'alley'. 'Boulevard', 'interstate', 'route', 'lane' are all kinds of roads.

'Street' could be a hypernym by the same reasoning, but as a native speaker, it does not feel like a generalization as much as 'road' does.

The hypernym for these, whether it is 'road' or 'thoroughfare' or something else, is not the same as a word for road names, that is, the things we attach to the name of a road when we say "Go two blocks, turn left at X". These are called odonyms (looked it up just now in Street or road names in Wikipedia). At the end they give a list of such names/odonyms which names you'll notice are not all acceptable as a kind of road (despite the fact that it is acceptable as the name of a road, e.g. 'close', 'mews', 'gate' passage', 'trail').

How about thoroughfare?

Says WordNet:

[bloom@cat-in-the-hat ~]$ wn avenue -hypen

Synonyms/Hypernyms (Ordered by Estimated Frequency) of noun avenue


2 senses of avenue                                                      

[Skipping sense 1 because it's not the word sense we're interested in.]

Sense 2
avenue, boulevard
       => street
           => thoroughfare
               => road, route
                   => way
                       => artifact, artefact
                           => whole, unit
                               => object, physical object
                                   => physical entity
                                       => entity

I'd say these ones:

• Route;
• Way;
• Path.