First let me caveat below with the fact that I'm not a professional.
I've never seen an install where a built-in tub was sitting on tile. That doesn't mean they're not out there.
As far as the advantage regarding the caulk and grout.. if the tub is causing settling of the underlayment, which would be a possible cause of your caulk and grout problems, I don't really see that tile would solve this. The tile will settle right along with the underlayment and if the settling isn't perfectly distributed, and it won't be, then the tile laid on top is almost certainly going to break up at grout joints.
Not even addressing the possibility that the underlayment and framing below the tub has moisture damage causing it to lose structural integrity. If that's the case you really need to fix that before you do anything else.
Another obvious issue is extra cost. Why pay to lay decorative tile where no one will ever see it? If it is in fact the case that adding a layer of ceramic under the tub provides added support and resistance to settling, and any pro would know this, then it seems to me that you can at least use a cheaper tile under the tub and have the tile types meet underneath the tub where they will be invisible. It also seems that a larger tile size would provide better support than smaller ones that you are most likely tiling the floor with. Fewer joints.
One other thing to consider I guess is that fact that if your tub is currently causing the underlayment to settle, then the weight of a tub plus the tile will make it settle even more. Perhaps actually, that's the key to all this. The weight of the tile causes the underlayment to settle as far as it's going to and so the additional weight of the water and person in the tub has a much smaller effect on settling than it would if that tile weight were absent. Again, a pro would immediately know this.
hope this helps... read the first line ml