Post History

Your question is about a particular subclass of German verbs, namely _weak verbs_.  Weak verbs, along with their conjugation, are a Proto-Germanic invention.  Proto-Germanic isn't an attested language itself, but every attested Germanic languages contains some reflection of the original Präteritum marker (suffix), and the most widely preserved feature of that marker is that it included a dental consonant.  [Its origin is uncertain](https://en.wikipedia.org/wiki/Germanic_weak_verb#Origin_of_dental_suffix), and the available theories do not seem to answer your question; quite the contrary, the dental consonant was present in all grammatical persons and it didn't disappear from 3rd person singular in any Germanic language (where we can tell it apart from the adjacent morphemes, at least).

Your answer automatically assumes that the right tense to look at for the "general" conjugation is the present tense.  We can, however, tentatively declare the Präteritum the source of our inspiration and then your question turns into this one:

If the 1st and 3rd person singular ending is, generally speaking, -e, where does the 3rd person singular present tense -t come from?  (That's the _other_ dental consonant missing from the hypothetical "*arbeitetet".)  It turns out that this dental consonant, only found in 3rd person singular present tense, was present already in Proto-Germanic.  It was present even in Proto-Indo-European (before the innovation described in the first paragraph of this answer), but there's a major twist: it existed also in 3rd person singular past tense in PIE, unlike in Proto-Germanic.

[Wikipedia](https://en.wikipedia.org/wiki/Proto-Indo-European_verbs#Proposed_endings) offers a key insight (confirming our tentative premise above): *"[...]from a diachronic perspective, the [past tense or tenseless] endings were actually the more basic ones, while the [present tense] endings were formed from them by adding a suffix, originally -i in the active voice and -r in the middle voice."*

My interpretation therefore is that this present tense -i added in PIE times somehow "protected" the 3rd person sg marker, a dental consonant, from disappearing during the evolution of Proto-Germanic; and this was before weak verbs were even invented.  That dental consonant became a fricative in Proto-Germanic, before it shifted back to a plosive in German.