If they did not like the dognapping caper and who he hangs out with, they would have cut him then.
They cut him the day that they found out about this additional larceny situation. There also was a question on how truthful he was about the situation. We don't know what he first gave to the team as a reason for the situation and later on his agent is doing his job and trying to spin this into a mysterious case of mistaken identity.
So did that person also supply the authorities with his SSN, since there is a record of a Rodney Whitehead being booked along with his SSN? So some random imposter drops his name and vitals when arrested? Highly dubious. Believing his story requires a big leap of faith and the belief in a conspiracy to frame him. That along with this dog thing and his behavior really provides a picture that it is easy to assume why they cut him. They were simply worn out by a marginal player bringing unnecessary drama.
If he was on the roster, he would get preference as an incumbent. That makes it harder for any other receiver to give the team a reason to give him more reps. He was such a specialized role, Switzer would have been his primary challenger. And in the meantime, he would be commanding reps over say a player like Noah Brown. It would really not be a case where they "couldn't handle it". This simply clears the path for the team to look at other options, rather than waste time with a problematic player.