Slight clarification to the FAQ: You don't get the recruiting money for those open slots until a full season later unless you fill them in the meantime.
So, if you cut a player at the end of Season 52, you get no money for the open slot in the ensuing recruiting cycle. But if that slot is still open in the NEXT cycle, you get the appropriate budget dollars.
In addition, I haven't heard of anyone losing a player because they cut someone, and the loyalty hit hasn't hurt me that much when I cut players, though I don't tend to do it very often.