I have a chimney just like that, which I built 20 years ago. Fireplace pictured at left.
My crown is identical to yours. I can't remember just how I did it, I am a log builder not a mason, but I think I just used regular mortar.
My crown has no cracks in it.
Yours is definitely toast. No doubt that mortar crown can be removed without damaging the clay flue tile. You could, for example, get a drill with a 3/8 masonry bit and drill holes an inch apart, all around that clay tile. Drill other holes as necessary and get a chisel and hammer and you can cleanly remove that crown.
I didn't use wire mesh. Maybe I had beginner's luck.
ps Your experience is why, when my fiancee and I buy a rental house, I do the inspection. When I discover a flaw, we just hit the seller up with the cost of repair. In your case I would have said "$400 to repair chimney crown."
They give us the discount on the price, and then after we buy the house we fix it ourselves.
The seller has no motivation to do a good job, in fact, they have a motivation to rig it and go ahead with the sale, once the deal is closed the problem is dumped on the new owner.