"...So I don’t understand why rangefinding is so critical? ..."
With all the variables, you'll miss a few even if you get the range estimate exact. If it's not exact, you'll miss more.
"...why laser rangefinders aren’t allowed?..."
Why? - who knows for sure? Just accept it. But FT is a game about rangefinding, and focus rangefinding is the easiest and most common method that is allowed, so that's what they do.
"...What I don’t understand is what rangefinding has to do with being a good shot?..."
There is more to the game than being a good shot or a good marksman.
Focus rangefinding is a particular skill that is somewhat unique to FT. It can also be applied to hunting/pesting out to 50 or 60 yards. Beyond that, it's of little use. Only a few companies makes scopes specifically designed for rangefinding in FT. And they are typically very pricey. Though you can get by with other scopes if you spend the time to learn/calibrate the scope.
For farther distances, a laser range finder is the ticket.
Also "ranging by eye" can be useful if you know what you are looking at. When you are familiar with the size of a target (or prey), you start to recognize the distance with regards to how much of the FOV that the target is covering. When you have a stadia type reticle (mil-dot, moa, etc.), it is even more obvious. Besides instinctive/subconscious "ranging by eye", you can do conscious "ranging by eye", where you learn more exact target sizes & stadia spacing in order to get even better range estimates. It's sometimes called bracketing and it can be useful at any distance (not just 10-55yds) if you know your target/prey and scope. It is a skill that a few long range rifle shooters still bother to learn (though most use a laser these days).
Bracketing in FT was more beneficial when I was using 12x scopes in Hunter Division. Now that I'm using 16x (in Hunter division) and 24x otherwise, I rely a little more on focus range finding. Though I'll still bracket in FT when a convenient feature presents itself.