I think he will stay. The FA Cup display might have shifted the odds in his favor, not because he won, but because of how he won it.
He could have easily parked the bus and hit them on the counter, but instead he chose to approach them like he did Barca last year and it payed off.
With this I think he might have more sway when he meets the board and tell them "See, I told you it was the injuries. Once I got a fully fit XI, we can go toe to toe with City"
On the other hand I don't see our board sacking him simply because of the alternatives. Poch despite his strong finish, did what you'd expect him to do. De Zebri is just another Potter and McKenna hasn't even managed a PL game yet. Tuchel might seem like an improvement, but his CL aside, what exactly did he do that everyone expects him to be an instant upgrade on ETH? Di Matteo won it as well, and Grant came runner up to the CL and PL in the same season.
If someone like Pep or Klopp was available, I'd understand why our board would sack ETH.(kind of like Mourinho looked like a vast improvement on LVG at the time), but currently any manager is a side move at best.
Considering that he won 2 cup finals, while marginally losing another one, and having improved so many of our younger players, based on this alone I think he deserves another season. Questions do have to be asked of his tactics though. The holes in the middle and the insane injury list have to be at the top of the queue.