Think about a FEL as a see-saw, front axle as the pivot point, anything behind front axle (even loaded tires) = ballast. Ballast boxes or rear implements provide extra leverage - tractor frame carries the weight.
Loaded tires - extra weight resting on ground, applied to frame if FEL tries to lift rear axle.
Two different pivot points.
1. Avoiding the back wheels coming off the ground. Any weight behind the front axles helps with this - for the rear wheels to lift anything behind the front axle must also be lifted. Loaded tires help with this, heavier tractor helps, ballast helps, an operator who drinks more beer helps
2. Reducing the load on the front axle. Loads on the loader put a lot of pressure on the front axle, and the front axle is usually smaller than the rear. Any weight behind the rear wheels has the effect of using the rear axle as a pivot, and putting upward pressure on the front axle - i.e removing weight from the front axle. Only things behind the rear wheels do this - so a ballast box or rear implement has the additional benefit of reducing loading on the front axle.
For this reason, I prefer a ballast box or a rear implement to loaded tires. I also mow with my machine, and I'd rather not have the weight on the lawn when I don't need it - a ballast box can be removed, loaded tires not so easily.
Having said that, loaded tires are very good for hill stability when going across the hill - which a ballast box does little for. Horses for courses. I guess spacers also help with that, so it's possible to get similar benefit without loaded tires.