There are many (duplicate) questions about the acceptance, popularity and history of singular they (and their, them and themself) around here, it even got a tag of its own. If I didn’t miss anything, however, the proper verb form hasn’t been questioned yet.
As we all know, English third person singular pronouns (it, she, he and one, +body), names (Alice, Bob, …) and nouns (student, teacher, …) demand the +(e)s suffix be added to the finite verb form in simple present, where some auxiliary verbs have a “special” form (is < *bes / *ares and has < *haves). All other subjects don’t, including plural third person pronoun they.
When the plural you replaced thou (with thee, thy / thine), the other marked verb form that had remained in English – i.e. suffix +(e)st or +t – vanished, too. The first and second persons only ever appear as pronouns (I, we; you), not names or nouns, so there was no strong inclination to keep the verbal inflection.
The second person precedent would suggest that singular they be used with uninflected verb forms which is how it’s usually encountered in the wild. Assuming that they someday replaces he and she (and maybe it) it would lead to disagreement with the words the pronoun stands in for:
- Alice goes to her place by herself.
- Bob goes to his place by himself.
- Alice and Bob go to their place by themselves. – (common)
- Alice and Bob go to their places by themselves. – (separate)
- She goes to her place by herself.
- He goes to his place by himself.
- They go to their place by themselves.
- They go to their places by themselves.
- They ?goes to their place by themself.
- ditto
- They go to their place by themselves.
- They go to their places by themselves.
- They ?go to their place by ?themselves.
- ditto
- They go to their place by themselves.
- They go to their places by themselves.
So why doesn’t singular they afford s on present-tense verbs like all other third person singular subjects do?
I learned about singular they only long after I had been taught English as a second language in school. That’s why it’s still a conscious decision to use it and hence I could easily adapt to use s forms with it, but would that sound and look funny / strange / wrong to native speakers?