It is not in a nature of Sith to plan for "true" successor, i.e. Apprentice that would go stronger and kill them. Exception could be Darth Bane in Legends, but in Canon both Darth Sidious and Darth Plagueis planned to live and to rule as much as they can. There is description of dynamics between Master and Apprentice in novel Lords of the Sith. Duty of the Master is to show only strength to Apprentice. Duty of Apprentice is to be unquestionably obedient, until the moment he is not. Then their confrontation would decide who is Sith Master.
And then, there is the question of Anakin Skywalker, Chosen One of the Force. Both Jedi and Sith were afraid of his full potential, because it was highest in the Galaxy. Jedi were reluctant to train him, and Palpatine famously said "You will not stop me. Darth Vader will become more powerful than either of us. " Neither Jedi or Sith could afford to ignore him, so they tried to persuade him to join their side and control him. I'm certain Palpatine would not just roll up and die, even if Anakin was not crippled in a duel against Obi-Wan. But again, he could not fully rule the Galaxy with such powerful being against him.
What about Dooku ? Dooku certainly was not stupid. He understood very well that Palpatine is younger and stronger in Force then he is. He also understood that it was politically expedient to have him at the head of CIS (well-known ex-Jedi master) . His mistake was belief that Palpatine could still have use of him when he was captured, as was their original plan for attack on Coruscant. Dooku was certainly aware of Anakin's potential, but considered him too young and raw to be true rival at the end of Clone War - his biggest mistake. But is worth of mentioning that Dooku did have some schemes of his own against Palpatine. for example, there are hints that he repeatedly tried to turn Obi-Wan Kenobi to his side, so that they together could go against Palpatine. Also, Palpatine ordered him to kill his assassin Asajj Ventress when she become too powerful. This possibly shows that Palpatine was afraid that two of them would plot to remove him.
In a sense, Dooku's apprenticeship with Palpatine was always temporary affair, it was always matter of moment when either of two would go against other one. Dooku had his chance when he held Palpatine "captive" aboard his ship. He didn't use it, and that was end of him.