You can't even rely on this, on my car the default behaviour is that the middle of the accelerator pedals range of motion is the "sweet spot", and in fact flooring it will trigger the hill climb assist and cause the automatic transmission to go down a gear.

Typically there is extra resistance at the downshift position, and you already have full throttle in the current gear before engaging it.

I haven't owned a car with them, but I thought those modes were to control when the gear shifts happen. Sport will optimize acceleration while eco will change up sooner to save fuel

My car has them, but it is a hybrid with planetary gear, so would not make sense for it.

I hope Sport mode changes both rpm limit and pedal response.