Yes, this construction is idiomatic French to describe a causal chain of events.

It is one of the few cases where a comma can be used before et :

Plus on sait, plus on est persuadé qu’on ne sait rien, et plus on est indigné d’avoir pris en vain tant de peine. Augustin Calmet, 1713

Et should only appear before the last statement if the goal is to chain all of them. Otherwise, that would break the chain.