Why can't wing-mounted spoilers be used to steepen approaches?
They can. The Airbus A318 and Embraer E190 both do this in steep approach mode. You can see spoilers extended in this video of a A318 landing and a similar approach in an E190 at London City Airport.

Going back further, the Lockheed L-1011 did this with a system called Direct Lift Control. When landing flaps were selected, spoilers raised a set amount. Small movements on the yoke raised or lowered the spoilers instead of adjusting the elevators. This made for a very smooth approach and landing.

The common point is that these aircraft use electronics for spoiler control. It's too complicated to do this in a cable flight control system. You need to have the system retract the spoilers quickly in cases like go-around and stalls. As I recall, steep approach mode on the A318 activates a different set of gains on the flight control loops for better handling qualities in that configuration.

Finally, a number of fighters have the same system, and some glider pilots do this manually.