In each of the cases where someone says they went out to a field and found large cents, buttons, coins, etc.... it is inevitably because something was there. Not just "random farm fields". By "random farm fields" (cultivated ag land) that had never had anything there (no houses, no stage stops, no camps, no river-crossings, etc...) you would be relying on random field-worker losses. To THAT extent, no, random fumble finger field hand losses are not prolific enough in our young country, to be worth just wandering around to find.
But in cases where something WAS there, then that's different. And sure: if the field in question had some reason to suspect that (crockery, brick, oyster shell, glass, etc...) then by all means, maybe something was there.
Here's an example of the demographics of just regular field worker losses: Where I'm at, agriculture started about the turn-of-the-century (before that was just cattle primarily). And there's a particular furroughed field that had a European influence factor (a contact-era indian rancheria) that existed from the 1790s to the mid 1830s. After the 1830s, there was no more human influence/presence. Till, of course, ag started there in the teens or whatever. And even though we hunt the place for spanish reales and buttons, yet we have noticed that over the years, yes, we find an occasional nuisance wheatie, one time a buffalo, and another time a silver washington. Even a few memorials and a clad dime. But in no way shape or form would it ever have been worth hunting this field, for "that buffalo" or "that silver washington" in the amount of acreage the site covers. They're just needles in hay-stacks. It's only because of the earlier village thingy that we're even hunting there.