I never understood why people care about torque from an engine when it's going to be connected to a gearbox that can convert the torque to whatever you want anyway. So why is torque a more important spec than power for an engine?

It's usually because there's an implicit "everything else the same" - and so if you build a car engine entirely focusing on peak horsepower you might end up with a dog because the curve isn't well suited to the rest of the design, whereas if you build it for maximum low-end torque you get a kick in the pants (and then strip all the gears or snap an axle).

Especially since people often consider "horsepower" to be things like turbos, etc that have their own spool-up requirements and result in inability to launch well.

They are related by the shaft speed anyway.

The reason that a lot of applications consider torque is that's more efficient to operate near the required shaft speed than it is to run faster and gear down. You fight more inertia in the engine at higher speed, and you add rotating mass with your gearbox.

An F1 gearbox can convert your torque to HP, but the gearing may or may not survive very long. If, like marine or rail applications, you need a fairly constant output power that isn't brutal on the spinning bits of metal, you will optimize for high torque at relatively low RPM.