Provinces were the administrative building blocks of the Empire, with Italia (Italy) having a separate role until the 4th century, split in 11 regions, so, yes, they were absolutely critical since without provinces there is no Empire.

In essence this would be close to asking whether the states were critical in the success of the USA, of course they were, the question is to what degree and in what respect. Some of them were economically important, others as sources of recruitment, for certain trade goods and so on. What is it that interests you about the provinces? That could be an interesting question.

You need to rephrase your question, not sure what you're actually asking. Your question is like asking are the states critical in the success of the USA