So dispel the myth. What actually happened. You have said the show was an unexpected hit. Surely the reaction to such a surprise success isn't typically to find a new actor for the starring role.
Nothing happened. Eccleston was only ever contracted for one year. Everyone involved has repeatedly said this, but I know some fans prefer their conspiracy theories. The fact of the matter is that
nobody thought the show would be a hit, least of all the people working on it.
It's worth noting the BBC itself
admits to signing a hush deal in January 2005 for both parties to keep his departure under wraps. This was three months before "Rose" aired and
seven months before shooting began on the second series. The only reason this whole thing has entered into fan mythology is because someone leaked his departure the day after the first episode was transmitted. It was
supposed to be a shock ending for "The Parting of the Ways," and Russell T Davies still talks about the surprise being ruined five years later.
If you look at Eccleston's CV prior to Doctor Who this all makes sense.
On the other hand, believing Eccleston was booted in favor of Tennant not only requires us to disregard all the public statements to the contrary despite not a jot of evidence, but also requires us to assume Eccleston was unprofessionally waffling well beyond a deadline, that Davies unprofessionally sought a replacement behind his back and that Tennant is an unprofessional opportunist. And moreover that all of these people were abetted by a highly unprofessional and backstabbing BBC.
And just for the record, Davies never had final hiring power for the Doctor. He could never have signed Tennant without approval from the top brass within drama, and there's no way they would have risked alienating one of their most respected talents that way.