Can a successor in the caliphate also be a reviver (Mujaddid)?
The caliphate is the higher and lasting institution, while the reviver was a substitute for times when it was absent. In a wider sense, however, a successor whom God enables to fight particular errors and wrongs can also be a reviver.
There is a tradition according to which God raises a reviver for Islam at the start of every century. To answer the question, it helps to distinguish two terms. The caliphate is found in the Holy Qur’an, as the verse above shows, and is described there at length. The word reviver does not appear in the Holy Qur’an.
From this a conclusion can be drawn. The caliphate that the Holy Prophet Muhammad (saw) left behind was, by his own indication, not to continue for a long time. For the period after that a substitute was needed, and that substitute was the revivers. Where the original is present, no substitute is needed. Whoever has water does not look for a makeshift. That is why it is hard to imagine the institution of revivers working alongside an existing caliphate.
For the last days the coming of the Messiah is foretold, and after him a caliphate on the pattern of prophethood is to arise again. In that context no further revivers are mentioned. This too supports the idea that the reviver is the substitute for times without a caliphate.
There is, however, a second and wider meaning of the word. The greatest of all revivers was the Holy Prophet Muhammad (saw) himself. He was sent for a renewal without limits, in every field and for all times. In this sense anyone appointed for a particular reform is a reviver. Even in the time of the caliphate, errors of belief and wrongs in practice arise. When God gives a successor the opportunity to fight them, he is in this sense a reviver as well.
Throughout, the caliphate remains the higher, lasting and fundamental institution. Renewal in the narrower sense is subordinate to it.