This is a frequent disconnect.
It really depends on the scale of the business. In some large ones, there is always a person working to handle “must be fixed now” problems. If you are off shift, then it is the person on shift’s task.
In smaller operations with 24 hour operation, and infrequent “must be fixed now” problems, there will be workers whose job includes being on call. Sometimes the on call duty rotates, sometimes it does not.
So if you are from a land of large companies, it may seem insane for your employer to ask something of you outside your shift. Likewise, when you are a little guy scrambling to grow, getting out of bed once in a while and handling a problem to preserve that growth is your best option.
In the situation in this post, I would hope that the critical “get up and fix this” messages come through a different system, or have a way to cut through a “do not disturb” so the employees can mute the mundane messages.
It sounds like you are in the first group. My company didn’t staff 24 hours, but used to pay people extra when they were the “get up and handle this” person. The size of the pay, and the frequency of the events was tuned so there was always a willing pool of people queueing to be the pager carrier.