Stiffness and flex relate to the same feature, the resistance of the structure to bending. Bending forces are mainly the twisting caused by pedalling hard, pushing the bars in one direction and the pedals in the other. Pedals, being stuck out to the side have leverage on the frame. Loaded touring bikes have , in addition, the side-to-side sway of the rear luggage rack.
You need the bike stiff enough to resist the bending force but any extra stiffness is wasted.
In the old days when tubes fitted into std sized lugs, you made tubes stiffer by increasing the tube-wall thickness. It is much more (weight) efficient to increase the tube diameter which is now the universal solution.
Aluminium should not be bent so is used in fat tubes. Any material in fat tubing (steel, carbon, titanium) will be stiff. Generally, steel, ti and carbon are used in smaller tube diameters to increase comfort (but not always). Modern aluminium bike designs are usually much more subtle than the original fat-tube designs and are more comfortable.
MTB vs Cyclocross. CX bikes are much lighter in weight and not as strong so big stunts are generally avoided. They also have higher gears since courses avoid major climbs or riders carry the bike and run.
Competition CX bikes are totally stripped down. General purpose CX bikes usually have threaded eyelets for everyday things like rack and fenders. They are good all-rounders.