3rd person singular 's' does not derive as a phonological modification from Early Modern English or Middle English 'eth', but from Old Norse, the dialects spoken by Scandinavian invaders/settlers whose language merged with Old English in the Danelaw and produced major lexical, grammatical and phonological change.*
This merging process took place between 900 and 1200 AD, by which latter date the Danelaw and separate Anglo-Danish identity had long disappeared. The process took a dual form: in some areas all persons uniformly acquired final 's' to the root of the verb and, in others, the several OE endings simply disappeared, resulting in so-called zero endings.
Gradual modification and standardisation, from the 17th century onwards, produced the modern 3rd person singular of the simple present tense and zero endings for all other persons. The entire dual forms still exist for all persons in non-standard English ('was' and 'were' are also examples) with varying levels of 'prestige' i.e: social acceptability, but, like bad girls, they've gone everywhere. They are habitual speech forms wherever English is spoken by native speakers.
Sez you? I hear you cry. Well, I guess it don't matter where because you pays your money and you takes your choice.
*A phonological modification of final 's' to final 'r' occurred in Old to Middle Norse (eg: Old Icelandic) during this period, but did so in Scandinavia and Iceland and seems to have been independent of the ongoing linguistic situation in Britain. Some experts maintain that the 's' ending for all persons possibly originated in a supposed Scandinavian difficulty in pronouncing the 'th' endings of the OE verb paradigm. I consider this unlikely because ON had both voiced and voiceless 'th' endings, as did all other Germanic languages of the period.