Fixing cracked bathtub with hot glue

I'm not familiar with the second approach but, as you say, it would address the root issue of the tray flexing when in use and should prevent the problem recurring. It also does seem to be rather radical.

The main drawback I can see is that you are causing more "damage" to the tray and therefore increasing the risk of the repair failing.

I'd ask the company if you can contact (or be contacted by) someone else who's had this done so you can ask them what they thought of the procedure. Ideally you'd be able to see the end result and check out for yourself before proceeding. If the firm is reputable they should have no problem with this. In fact refusing this would be a sign that the firm wasn't as reliable as they made out.

In most cases, a silicone based glue would adhere fairly well to most tiles. On smooth tiles it can be removed with a razor blade. On rougher tiles, a blade and a wire brush can get it off. The tiles need to be thoroughly cleaned before using (as with any glue).

This is not as strong a bond as epoxy or construction adhesive, as suggested by Chris Cudmore, but is more easily reversed if necessary.

Because it is not as strong, I would limit it to lighter items, like the soap holder you describe. It is definitely not suitable for a grab bar or other attachment that needs certain hold. Safety items need to be attached to structure, not glued on.