A "mechanic" in 18th century English parlance simply referred to a manual laborer. It was sometimes used in reference to a tradesman of some kind, but more frequently it meant an unskilled worker or a day laborer. Here, for example, is Thomas Paine using it in this context, and another from Arthur Young - "down to the mechanic who fhould attend to the labour of his hands."

Check out this guide to 18th century vocabulary (search for "mechanic"), and the etymological dictionary entry on "mechanic." The word didn't pick up the context of working with machines until the automobile became widespread. The trope of the "mechanic" - meaning the simple worker - writing letters was used often by authors to frame political treatises (Crevecoeur's Letters from an American Farmer is a similar idea, just with a slightly different context).

I'm at home and not among my resources in my office (and at 11:30pm on New Years Eve, it's not happening. Sorry!) But a "mechanic" refers to a skilled worker who builds and repairs machinery. Think farming implements, guns, weaving materials, stills, those sorts of things. The kind of equipment that takes specialized equipment to build and specialized education to learn how to make.

It's a little out of my specialization, but I can't imagine too many people would be eligible to vote solely because they were a mechanic by trade. I would think there would be several people manufacturing farming implements and grist mills in 1770s Georgia, maybe a few people who build and repair firearms. If you're interested and if I remember, I can do some digging on the 2nd and see what turns up for a more concrete example of what an 18th century mechanic did for a living.