Oh. Oh my.

I'm a firefighter.

First of all, you risked not only your own life, but by entering a stairwell, had there been a real fire, you may have risked the lives of others by moving smoke from where is was to where it wasn't.

Hotels in the USA have a number of critical safety features, including 2+ hour rated doors, in-room sprinklers and floor-by-floor fire communications systems.

If you're below the fire floor, you should stay in the room, with the door closed. Get dressed, get your room key and a wet towell in case there's an "auto-exposure" situation where the fire moves DOWN (almost impossible).

If you're on or above the fire floor, for the most part, your room is going to be the most secure place to stay, in the vast majority of cases. Smoke is your biggest problem, and you can hold smoke out of the room with a wet towel at the base of the door.

Sprinklers will also kick on, not to put out the fire, but to reduce the smoke levels.

FWIW - there has never been a fire fatality in a properly sprinklered building. Not one.

As far as the 10 minute fire response time, the national goal (NFPA 1071) is 6 minutes, which, if you are a firefighter, you know is often unrealistic due to staffing issues and funding. And - to be honest - an automatic alarm call is - 99.98% of the time - a false alarm, so you might not see the kind of huge response you'd expect. Especially with a "chronic" alarm system, as this one sounds.