The entire point of the Rule of Two was to prevent a bunch of Sith running around squabbling over power, hindering their overall strength, and to stay hidden from the Jedi. The master taught an apprentice as much as possible and when the apprentice thought he/she was strong enough would overthrow the master or be killed attempting it. Both parties knew the stakes, so it was obvious they would seek out replacements.
The thing that, in my opinion, stays consistent with only two is that even though they sought other apprentices, none were ever full Sith, until the master or apprentice was dead. Despite other potential replacements, there were only ever 2 Sith at any time.
Palpatine started looking at Anakin to replace Dooku bc he knew Dooku was so old already that he'd never be able to continue the line or train someone fully. He used him to consolidate the separatist movement. Once he felt that Anakin could be turned, he orchestrated events so Dooku would be dead and he could focus his efforts on Anakin.
Meanwhile, Dooku starts thinking about overthrowing Palpatine, and picks Savage to train. Easier to kill the master with some help than it is by yourself. Too bad for Dooku, Maul shows back, gets Savage to side with him and tries to kill Dooku. So much for Dooku overthrowing Palpatine. Had Maul beaten Dooku, either he and Savage kill Palpatine and become the 2, or Palpatine kills them both and continues his plans for Anakin.
Once Anakin turns, he immediately goes and gets his arms and legs cut off, which kind of puts a ceiling on how powerful he can become. Too bad, there's no one left to turn, so Palpatine keeps him around bc he's still pretty much the most powerful force user alive.
But what's this? Luke Skywalker hits the scene and now Palpatine has a suitable replacement. Get rid of Vader, teach Luke...still 2. Vader had the same plan. Turn Luke, join up to kill the emperor...still 2.
