Consider this conversation: WOMAN: The cat too. MAN: The cat too what? WOMAN: The cat too goes. Treats the phrase "the cat too" as a subject, the point of the sentence, not the cat "getting to go" but the cat being singled out as if on a list. Again, I don't think this would be the normal way of talking except for emphasis or poetic reasons. I could be wrong. My mom taught English and my dad was a Brit Lit major. Both acted semi-proffesionaly. They corrected my English as long as they lived. Saying "the cat too goes" sounds like a drama queen, almost obnoxious, if not technically wrong. Which it might be.
I just spent half an hour with three different grammar books and I searched three different web pages, but I can't find anything on this subject. :) However, you're explanation seems pretty valid. I guess Duo didn't except "the cat too goes" because it is extremely uncommon.
Also, correct me if I'm wrong, but wouldn't it be more correct to write that with a comma before and after "too"? Like "The cat, too, goes"?
идти́ (idtí) [ɪˈtʲːi] impf (perfective пойти́) (concrete verbs) "to go; to walk" From Old East Slavic ити (iti), from Proto-Slavic *jьti (“to go”), from Proto-Balto-Slavic *eitei, from Proto-Indo-European *h₁éyti (“to go”). Cognates include Lithuanian eĩti (“to go, walk”), Latin eō (“to go”). The infinitive spelled with д (d) is attested since the 14th century, under influence of иду́ (idú), идёшь (idjóšʹ), etc.
As a curiosity, the Spanish verb "ir" ("to go") -as long as many other Romance equivalents- was built by mixing up multiple Latin verbs [the i- and y- forms from eo ("I go"), the v- forms from vado ("I go, walk"), and the f- forms from sum ("I am") or fugio ("I flee")], so "yo voy" ("I go"), "yo fui" ("I went"), "yo iré" ("I will go"), the i- and y- forms being cognate with идит.