I've been at a handful of airport hotels with very early breakfast available, e.g. one at SCL that started breakfast at 4 a.m.

Most do have coffee and tea available at all hours in the lobby. What I usually do is stop at a supermarket and buy a muffin and a piece of fruit the night before. Or just eat breakfast at the airport.

I'm more annoyed with business hotels that don't start breakfast until 7 a.m. - kind of rough if you have an 8 a.m. meeting.