From your first paragraph, I assume that each cycle of an individual universe contains different people, doing different things, from those in preceding and following cycles.

We seem to agree that "the term 'infinite number of times' is not correct", but for different reasons.

If my understanding is correct, you regard the past and future of eternity as limitless. You seem to have done your own bit of "renormalisation" here, effectively getting rid of the inevitable infinity and replacing it with two limitless periods of time.

I would maintain that the term 'infinite number of times' is not correct because it implies that infinity/eternity can be measured in terms of time.

You say "the infinity in time cannot be removed completely", as though the addition of the word "completely" means that you have removed enough of it to be able to discount it. (more renormalisation) I would maintain that the infinity must either be in your calculations, or not. You can't do away with part of it and ignore the rest.

There never was nothing.