Why do any of the other variables matter if the market is collectively fighting between "overpriced and underpriced" on premium/implied volatility?

The Black-Scholes equation describes the unconscious biases that influence the price of options. It is able to beautifully separate the one quantity that every trader prices differently from the rest. That’s volatility.

All the other factors, time including, are the same for everyone.

You can find an IV that makes sense for a single option with invalid other parameters, but things will break down when you go to price other expirations / strikes.

When trading, you don't want to wait to see an "updated" IV, you would want to respond directly to changes in important and well understood parameters like underlying price.

They don't really matter. They are just things you look up in order to a get a number out for what the option costs in dollars.

Was BS actually of any practical use? It just seems to be a fantasy like most maths in economics.

Yes, it has some use. First of all, if you don't have a common model, it becomes impossible to talk about vol. So even if everyone uses their own model, they convert back into BS vol to talk about vol. Second, all models are wrong, but some models are useful. BS is a good starting point because it captures something that is relevant in option pricing, namely that uncertainty matters.