Since private sales are legal isn't it likely that the people who cannot pass a background check can just go the eBay route or the Craigslist route and get a gun? There is no obligation to do anything but verify they are 18 years old and live in the same state as the seller does.
Well, except that neither eBay nor Craigslist allow the sale of firearms, you are essentially correct.
All anyone who wants to buy a firearm needs to do is go to www.gunbroker.com or open your local Penny Saver and look in the Sporting Goods section. Private individuals are selling firearms every day.
So if a criminal wants to buy a used firearm at market prices all they need is cash and they can buy one, in most states, no questions asked.
The way around this is to have universal licensing and require that all sellers keep a record of the license information from the buyer for 10 years or so. This is what Illinois does. Sellers have an incentive to only sell to licensed buyers because if you sell to someone without a license there is a good chance that person is a criminal and will use that firearm in a crime, whereupon it may be traced back to the last legitimate owner.
However, what I have recommended is instead of an opt-in system like Illinois has, where only firearm owners apply for FOIDs, thus creating a registry of all firearm owners, they should have an opt-out system, where FOIDs are automatically issued to everyone who applies for a driver's license or state-issued ID, unless they opt out. By so doing, merely having an FOID does not guarantee that that person actually owns firearms. Thus firearm ownership anonymity is preserved, but you can also help make sure that private sales are only between authorized individuals.