When we look at the pair *-mén(i)/*-tén(i), the so-called plural *-n- appears to be only explainable as the product of analogy with early MIE 3pp ending *-éna before the time when *ta “that” (> PIE *to-) was employed to extend the 3rd person singular and plural endings. This termination may once have spread from the third person plural active to the rest of the plural terminations at an early date. QAR predicts accent on *e and the etymology of these endings is transparent indicating that *-mén(i) and *-tén(i) date to at least the Mid IE period before Syncope had yet to take effect.
Now, if we know that *-mén(i) and *-tén(i) are quite ancient, it follows that the accentuation of *-més and *-té can be explained as analogy with older *-mén and *-tén. However, the lack of *-i in primary *-més and *-té still begs an answer. Logically, whatever the source of these unextended endings, they must have once had no need for the indicative *-i. This may indicate a particular usage outside of the primary conjugation. I believe that a possible reason for this is that these latter pair of endings were taken directly from the independent oblique plural pronouns of the time: *mes and *te. MIE enclitic *mas regularly becomes *n̥s via Syncope, and was then later extended analogically as *nos by the time of PIE proper. MIE 2pp oblique *te however (*tei in the nominative case) was replaced in the meantime by an inanimate noun *yáuas “(the) group” (> early Late IE *yaus), thereby obscuring the ultimate source of later 2pp ending *-té.
If this is all correct, it's then probable to me that these alternative endings were first coined as early as the late Mid IE period and that dialectal replacement of *-méni and *-téni by *i-less, pronoun-derived alternatives *-més and *-té began to spread during the Late IE period.
Thus I think we now have a sensible solution to the reconstruction of the Old IE objective endings preceding the agglutination of “indicative” postclitic demonstrative *əi (> PIE *-i):
Furthermore, we may possibly reconstruct both the singular and plural independent pronoun forms for the first and second persons with greater depth:
|1st person||*məi (nom.)||*wəi (nom.)|
|*mə (enc.)||*məs (enc./obl.)|
|2nd person||*tau (nom.)||*təi (nom.)|
|*tʷə (enc.)||*tə (enc./obl.)|
And now everything in the 3000 years prior to PIE is explained...
... Or is it?! Alas, my work is never done. Happy Halloween, everyone!
 The basic root *yeu- "to bind, join together" is acknowledged in Mallory/Adams, The Oxford Introduction to Proto-Indo-European and the Proto-Indo-European World (2006), p.522 (see link). It is also deemed the underlying root of extended *yeu-g- with identical meaning which is the source of inanimate thematic *yugóm "a yoke" as expounded upon by Szemerényi, Introduction to Indo-European Linguistics (1999), p.272, fn.10 (see link).