There is hardly any team in the league that can not sign a particular player, regardless of its alleged cap status, if it really wants the player. Some signings do take more work than others. That's the downside to pushing these payments out. Cut or trade a player, and all that uncounted money suddenly counts immediately. But there's a lot more of the other going on.
Teams can sign the players they want.
Because teams really can afford what they want. Mr. Cap is just a convenient scapegoat.