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