JT,
Not sure why a team doesn't take a player than offers FA. Sometimes it's because the scout who was following you strongly believes in you, or for reason you just slipped off. Or players have asked for releases and they need to fill a spot, hurt players. It also gives the player an opportunity for a bit more than a later round would pay. Understand that many teams rely on later rounds to help fill their milb team needs, which is essentially to field teams.
Having an agent is expensive, but there are some guys out there who will definetly help a player find an option. A lot of it is who you know. Most players drafted in later rounds are essentilly warned to stay away from advisor/agents who promise them things because they have no money to fulfill those promises. IMO an advisors job is to provide you enough help and support beforehand so that you will get drafted on draft day get some money and pay for his services.
As far as not getting drafted, it's up to the player, does he want to move on or make a run for it by playing ball that does not pay you a salary (Indy Ball).
BTW, Cards drafted a player from Rowan(?) and a player from Belmont Abbey, so you do have to know some teams are everywhere looking for the best players.