Grout seals nothing and is extremely porous. Even “sealed” grout does nothing to waterproof a shower, it just helps the grout from absorbing stains. You need a robust waterproofing system that covers the walls and floors and overlaps at the seams (Kerdi is a good one, but must be installed correctly). The boards need to be replaced because they are rotted. Your waterproofing system should run up the walls as well, so the tile needs to be removed. It’s lucky that no one fell through the floor while showering.

I agree with AB normal above. Please take some time to look at some of the YouTube videos about shower tiling and waterproofing fails. They are very instructive about how bad a situation like what you have can be.

the supporting joists need to be examined for rot or structural failure and replaced sooner than later otherwise you can have more on your hands than a leaky shower.