It seems to be more of a debate of tires filled with methanol/water or beet juice vs. iron weights any more as calcium chloride has certainly decreased in popularity. The main drawbacks to filled tires are that it can be some level of mess if you get a leak (most notably with beet juice), it's harder to fine-tune ballast, and that it interferes with sidewall flexion on radials. The latter really seems to be why in new larger tractors filled tires are going out of favor- they generally come with radials and the tire reps don't want the customers to see there's little difference in performance between a fluid-filled radial and a cheaper fluid-filled bias tire. The advantages are that you can often add more weight with liquid than you can with just iron, it's much less expensive (especially with alcohol), and you can always add cast iron weights to filled tires to get even more ballast if needed. My tractor's rear tires are filled with water/methanol as they're bias tires, it adds more weight than I could with cast iron weights, I have a permanently attached loader and won't be needing to lessen ballast, if I get a leak it would be a service call no matter if I used cast or liquid ballast, and it was very inexpensive. If I ever need to add more weight, I can always add cast. Versus beet juice, alcohol is a lot less messy and much less expensive which is why I picked it.
As an aside, we only had one filled tire ever get a flat in a tractor growing up. It took a service call because even if the tire was air-filled and had no iron on it at all, it would have still weighed about 500 pounds and would not have been practical to remove and haul into the tire shop. The leaking fluid (which was nearly certainly calcium chloride) did nothing more than make a water puddle in the dirt floor of the machine shed. We did have a number of fronts go flat, which were/are never filled with fluid.