I understand the concerns about using dish soap if it tends to dry your hands. That hasn't been a probelm for me, but I do have a child with OCD who tends to wash more than necessary, so for a while I switched from the antibacterial dish soap to a dish soap that was not antibacterial. It was gentler on the hands, and kept my child's hands from cracking (and sometimes bleeding) as they had with the antibacterial soap. Since then, I've diverted my child from constant hand washing to using hand sanitizer in between a normal amount of hand washing, which has solved the problem. Now I can use whatever kind of dish soap I want. But if it doesn't bother your hands, dish soap is a perfectly good hand soap. (My grandmother even used dish soap for shampoo, which I do occasionally for a super cleansing of all hair products from my hair, with no discernible harm.)

I love my Never MT. We built a new house and put one in each bathroom, where I use hand soap(softsoap) and in the kitchen where I use dish soap. The only problem I've had is that in one bathroom, I had to replace the Never MT...it leaked at the valve. This has happened twice. But it's still worth it. Actually the first one was replaced by the manufacturer after 6 months use.

Mask around the good tiles. First use a grout saw to remove all the grout. The key is to chisel into the soap dish, so you don't chip the adjacent tiles. Wear gloves and eye protection, it's dusty also. Then you can give them a gentle but firm 'wack' with a rubber mallet. Some will break off. Then using a hammer, and a 1/2" chisel, start chipping about a 1/2" from the edge. A pro would remove the grout and probably use and angle grinder to cut an 'X" or star pattern across the soap dish.

I'd score it with a carbide blade in an oscillating tool then break it up by gently tapping with a chisel. First take out the grout surrounding it, then see if there is enough room to slip the oscillating tool blade between the soap dish and the wall tile. If you can score from the back side it should be easy to break it up without damaging the tile. Alternately take out both the dish and surrounding tiles and make it a "feature" by using a colored or patterned tile around the soap dish if you can't find matching replacement tiles. Some makers (eg American Olean) have replacement tiles that are still astonishingly good matches decades later for their standard tiles.