How is a question who’s edges are unique to the company but if you’re coming in to a situation like that one generally-solid approach is to accept that things will suck for a while, and might even be worse short-term, the prioritize chipping away at the underlying causes.
Something like reducing staff ticket time by 20%, then using that 20% for feedback, strategy, and structure. Some (maybe even most if you’re lucky) of the front-line staff will have enough experience and insight to be invaluable here (though it is likely they won’t have the language yet to express it in ways that make sense to management). As the company goes through the process of communication, discovery, awareness, planning, and execution (preferably with a tight feedback loop) some of the underlying causes will be addressed, the front-line pressure will ease off, the cascade effect will see ease-off in other departments as well, and that 20% can go down to 5% or even 0 (not recommended lol) which will further reduce the workload to give longer-lasting relief.
Then with staff time “out of the red” the company can start thinking about what they will do in a fire-free future.
Distributed prioritization seems like a problem; you can get priority inversion if you’re not careful.