mnlg:Complete the following sentence:

"Something or someone broke into my house last night and I won't rest until I catch ____!"

I believe a grammarian would probably say that the correct word to use is "him". Remember, in "correct" English there is no gender-neutral singular personal pronoun, and the masculine, as a kind of default, does what we want "them" to do now. In the past, "him", in this context, would be considered gender-neutral rather than masculine. Nowadays a shift is occuring, and "him" is nearly universally considered purely masculine, having been replaced by "them" in the spoken language. But it was not always so. And the written language still, for the most part, follows the older style. If the sentence example you gave were in a newspaper article, it would almost certainly end in "him" and not "them" (unless it was a verbatim quote from someone being interviewed).

Beginning several decades ago, some authors started reversing the standard masculine-default when writing, and use a feminine-default instead (i.e., using "she" instead of "he" when talking about people whose gender has not been established). In that case, the final "correct" word in your sentence would be "her".

mnlg:I have been taught (by a native speaker) that the most neutral answer for this is "them", even if the subject is, in theory, singular.

Should I reconsider?

No, I don't think you need to reconsider. Nearly all English speakers (well... at least Americans) would say "them" as well. Including me. But, as I said before, it's not considered "correct".

Eventually it no doubt will be, however. As much as they might like to think otherwise, grammarians don't actually make the rules. They just describe them. The actual rules are determined by usage, and the widespread usage indicates that "them" will one day be officially considered singular as well as plural.

Some time back I checked the online version of the OED (Oxford English Dictionary) regarding the singular neutral-gender use of "they". For anyone who doesn't know, the OED is considered a definitive work on the English language, and has extensive documentation of the first noted use of various words, historical examples of usage, etymology, etc. It's pretty much all you ever wanted to know and more about the history of English words.

If I remember correctly, the OED noted that use of "they" and "them" in the singular for gender-neutral usage dates back at least to the middle ages. Suffice it to say, this is not a modern usage at all. Presumably grammarians have been railing against it ever since then, but people see a need for it, so they have used it for hundreds of years.

I don't use it in formal writing but especially after reading about the historical use in the OED, I now unabashedly use they/them in the gender-neutral singular. Those grammarians can ^$#@$@# their %#@%@$es, as I have the weight of historical usage behind me.