Complete gut and redo. There is sure to be water damage. If you wait, there may be even more, and it could affect the neighbors.

I wouldn't be so anxious to demo everything. If the existing substrate is sound it could be made to accept a modern tile application.

"Then the tiles come down like domino's since water got behind the grout."

That's a complete gut job.

Remove tiles.

I'll leave the assessment up to you.

If the mud wall is indeed tight on the studs/lathe, not delaminated where you can push on the mud and it moves, etc, then clean it up.

You can sometimes assess delamination by lightly tapping all over the wall with a hammer. Assuming it;s all good?

Remove any thinset/grout/mud ridges that stand proud of the flat wall surface.

Assess the divots that need to be patched. If they are shallow, say 1/4" deep max, you can patch with thinset. If deeper, you can patch with two coats of thinset; a first pass to fill half the depth of the divot, the next day another pass to fill the remainder of the divot. Thinset can shrink when it cures, that's the reason you don't want to pile it on too thick.

If there is a lot of patching, you can get a better bond by using a bonding agent.

Any lumps or ridges on the surface will knock your tiles out of alignment. So get the wall flat. If need be, you can even skim coat the wall with a thin 1/8" thick veneer of thinset to give you a nice final surface to tile upon. Be careful in the wall-wall corners that you keep a nice vertical crease in the corner. It might help to skim coat the back wall. Let it cure overnight. Then skim coat the two end walls the next day.

If you want waterproofing, you can use a topical waterproof membrane like Hydroban or RedGard. Paint/roll/trowel it on. Two coats.

The tile over the membrane.

Then grout.

You are dealing with a cement mortar bed. You are the best judge of its soundness. Lightly tap over the entire surface and listen carefully. You will be able to hear places where the cement is unsound. It sounds like tapping on a shell. Do you see rust bleeding through in many places?

Old buildings never have straight plumb framing. Cement board will not make up for the irregularities the way a mortar bed will. The final tile installation will either look lousy or take the installer 3 times longer to make it look right.

If you do decide to tear out the mortar bed, I'd recommend having a professional re-do the mortar bed. Then you can DIY the waterproofing and tile setting. The professionally done mortar bed will make this job about 100 times easier for you.

There are some types who would set 4x4 tile with dabs of grout or plaster of paris, then grout (same day) get the $$ and skeedaddle. When that crap fails, it's like you say, a cascade effect.
Casey