Yes to both your questions.

Native speakers really do follow that “rule” as to whether being next to a vowel or a voiced consonant makes the -s suffix voiced as well, whereas being next to an unvoiced consonant makes the suffix also unvoiced.

And secondly, the same phonologic law is in operation when constructing a third-person singular verb.

You can also add a third class to that: forming possessives with apostrophe-s.

All three work the same soundwise.

Part I

The rule that Peter pointed out in comments is that it is voiced only in function words, not in others. (In fact, this is more of a law than a rule really, because it has no exception in English.)

The complete list, excluding derived terms based on words in this list, is:

than, that, the, thee, their, them, then, thence, there, these, they, thilk¹, thine, this, thither, those, thou, though, thus, thy, thyne, thyself.

Notice how those are all function words of one sort or another, not nouns or verbs.

Note that English also has a few words that begin with th where the h is silent, like Thomas and Thames. Those you just have to learn by rote. There are not very many of them, but they can be quite common.

Part II

The second part of the question asks whether there is a reason for these differences. We don’t know for sure, but function words tend not to be stressed, which tends to make them run together and experience greater assimilation with surrounding sounds. This might have caused the voicing to stick around there.

Wikipedia suggests:

In early Middle English times, a group of very common function words beginning with /θ/ (the, they, there, etc.) came to be pronounced with /ð/ instead of /θ/. Possibly this was a sandhi development; as these words are frequently found in unstressed positions they can sometimes appear to run on from the preceding word, which may have resulted in the dental fricative being treated as though it were word-internal.

In a comment, I note it is also rare to find a word than ends in voiced -th (without an e following it), with smooth and the verb mouth /maʊð/ being notable exceptions. And by rights, that one really ought to be spelled mouthe, like all the others (bathe, clothe, breathe, etc.). Voicing wasn’t phonemic at the ends of words, and happened only with a following inflectional vowel. Hence unvoiced in nouns house, wolf, bath but voiced in verbs house, wolve, bathe, even when not in the third-person singular -s form.

This is related to the intervocalic voicing described above. It is retained even when we have lost final e, whether just in pronunciation or in spelling as well.

Footnotes

1. Thilk is an archaic or dialectal demonstrative, which per the OED is also:

thick /ðɪk/ is in dialect use from Cornwall and Hants to Worcester and Hereford; and also in Pembroke, Glamorgan, and Wexford. In many parts it has also the form thicky, thickee, or thicka. It generally means ‘that’, but. . . .