As for who is responsible for that decision, it's a role the government generally fills. Can it be a bottleneck? Yes. But there are extreme bottlenecks now with hospitals already out of, or about to run out of, PPE and ventilators. Might there be mistakes? Sure, but again there are mistakes right now too, with market forces favoring those who can pay, have inside connections, etc., instead of those who need the resources. In short, the detriments you cite are a reality of any system, while their presence during a crisis can be mitigated at least a little, and help critical need be filled at least a little bit easier.
Don't let the desire for a "perfect" allocation system blind you to one that is merely "better" than what we currently have now with panic buying, hoarding, price gouging, etc.
Who is responsible for that decision? Do they become a bottleneck? Might they make mistakes? Are they susceptible to corruption? If so, it may be more wise to distribute that decision-making... via the market system.