The most fire sensitive part of a high building is the staircase. You want long burn time in fire but also have very little "fire load". (Amount of energy released from material that can burn in typical fire. Typically MJ/sqm.)
Steel staircase gets you 0 fire load but relatively short "burn time". Because steel gets soft at relatively low temperature and more importantly conducts heat very well.
Pure concrete stair case is better, but you get cracking that shortens the burn time.
Wood is surprizingly good for burn time, but you greatly increase the likelihood that the fire will start form the staircase. Which is really big no-no. Sometimes even burning a complete house (starting from somewhere else) is lot better than just burning the emergency exit.
The only really good option is reinforced concrete. The concrete insulates the steel, which provides ductility to the whole structure. With reinforced concrete staircase and stairwell, it's questionable how much you benefit from wooden load bearing structure.
I think it can be done. And it can be done really well too. But you may need to step away from typical architecture and step into something more modular.