Technically we would have to discuss each of the verbs separately, but I'll lump them all together.
These modal verbs are not just modal verbs. Some of them have a definition as a full verb, too, and others can be used as full verbs in some circumstances.
Strictly speaking, I would never understand "Ich kann Englisch." as a shortened form of "Ich kann Englisch sprechen.". It's plainly "Englisch können". Können is defined as beherrschen or wissen. As such, können would probably be translated as to know; thus; "Ich kann Englisch." is "I know English." and not "I can speak English.".
Können is kinda a replacement for beherrschen in that example. "Ich beherrsche Englisch." is certainly not a 'good German' sentence, but it's correct.
Anyhow, you can complete the sentence with an infinitive, see also canoonet. And sprechen is the word that usually is being implied when saying "Ich kann Englisch" as opposed to schreiben, lesen etc. You would have to mention those verbs, if you'd like to express another sense other than knowing a language. (Besides, that's the reason why your notion that you can always drop the main verb cannot be true.)
Looking at the other examples, in "Ich will nach Hause.", for example, gehen is not necessarily the implied action. Fahren, laufen, or even hüpfen are valid verbs. However, wollen is defined as beabsichtigen or anstreben. You're just saying that you want to get home, or rather that you would feel much better if you were at home.
"Ich muss zum Arzt." means that it's (more or less) urgent that you visit a doctor. Müssen implies that it is necessary to get somewhere; or away as in "Ich muss weg.".
None of these sentences is really a shortened form; though, a certain verb is often implicitly implied.
So, why doesn't work "Ich kann Auto."? The verb fahren is certainly the most likely verb. What else could you do with a car?
I don't really have a satisfying answer to that outside of mentioning that modal verbs used as full verbs are not completely universal. "Auto können" is kinda nonsense, as is "Auto beherrschen".
Funny enough, if someone asks you "Kannst du Auto fahren?", you're fine answering "Ja, kann ich." because then it's obviously an ellipse of "Ja, ich kann Auto fahren.".
I feel like my answer isn't complete and I hope others can elaborate more on the "why", but hopefully I could make clear that modal verbs are more than just modal verbs.