Why doesn’t the A320 family suffer from the same design constraints regarding engine placement/landing gear length as the Boeing 737 family?
So how does the A320 design address this? Longer landing gear with attachment points further outward on the wing [...]?

Yes, the A320 has a longer and therefore wider landing gear. You can find general aircraft dimensions in the Airplane Characteristics for Airport Planning documents.
Airbus A320-200: 7.59m wide landing gear

Boeing 737-800: 5.72m wide landing gear


The same document also lists the ground clearances at different points. At MTOW (max. takeoff weight) or MRW (max. ramp weight), the ground clearance
below the fuselage is
A320: 1.72m, 737-800: 1.30m
and below the engine nacelle it is
A320: 0.57m, 737-800: 0.48m

Sources: Airbus, Boeing