Ultimately, "un-paid" refs are getting jollies one way or another - prestige, freebies, whatever. Theoretically, they could be "fired" for failures, which is a fairly small threat. But not even that seems to be done, at least not openly and with reporting.
To me, besides the idea of "pro" reffing, there is an element of criminality that would not necessarily be addressed with full-time refs... Gambling rears its head, subsumed under the general term "Vegas". Big sports books try to even out their bets and live on the vig. That's not always possible, and it seems to be a near certainty that when the bets fall heavily one way, a crooked ref will be required to make sure the outcome goes the "right" way. I don't know about NFL refs being professionals, but anyone who has watched NFL games more than a couple of times has seen calls that are obviously wrong and either aren't or just can't be reviewed or are reviewed and not corrected. I don't see full-time refereeing keeping that from happening.
I suspect the DPI call in the Georgia-Texas game was a result of exactly that sort of thing. The guy who made the call was way late throwing his hanky, probably because he thought he needed to prevent Texas from scoring. Between the announcers carrying on about the totally shit call and the fans going nuts, the word came down from wherever to not push the issue.
Meanwhile... how much would it actually cost the SEC to have full-time refs for football?