Landing climb weight assumes AEO and landing configuration (flaps and gear extended) - think a hydraulic malfunction where gear and flaps are stuck, but you have to do a go-around.
I think the 2.1% approach climb gradient required is similar to the 2.4% requirement during 2nd segment after takeoff - you'll have sufficient performance to fly away, assuming no obstacles in your way. The go-around climb gradient ensures you can do this safely in IMC, and have sufficient obstacle clearance along the way, which is done by satisfying the MACG requirement from the instrument approach chart.