Specifically the rate of star formation has always been dubious, because there is such an enormous lag between star formation and the emergence of intelligent life (even if we turn out to be an extremely slow outlier), making it unsuitable for estimating the amount of civilizations active at a specific point in time.
My personal drake equation would include only factors that are based on the current state of the galaxy. But that's the Drake equation doing its job: making people think about cosmic odds. It doesn't need an update, because I argue it shouldn't be taken literally in the first place.
A few times a year, authors put out "new" estimations of the number of galactic civilizations, and to do that there is really not much alternative to chaining together these kinds of probabilities. While they sometimes may or more often may not use the actual Drake equation to do it, its value lies in the basic structure, not in its specific implementation.