Airbnb doesn't just mean traditional Bed&Breakfast, it is actually a much broader concept.
When I travel, food is a big part of my experience - I always go for local food. If there is no east breakfast place around the place I am staying, I just grab some simple food(fruit, dessert, snack, etc) the day before and just keep myself not hungry - and wait for a big lunch.
@Jan13 If you want to find a place serving breakfast, you can click on the amenity section, and click 'apply filter'.
All of my listings are located in the very 'trendy' area of one of the biggest cities in the world, basically everything is here.
For food, the most trendy restaurant street is right next to my place, most are awesome(unless they are great and popular, they cannot survive this street) so my guests can find anything from simple to 20 course, cheap to posh, local to fusion, 24/7.
Also a lot of my guests are on the younger side - they go out at night, come home late, wake up late, most skip breakfast anyway.
It is way better for both my guests and me to not include the breakfast and give cheaper rate.
If they want to cook themselve (usually longer term guests), they can most certainly do so - I provide kitchen and cooking tools, there is a traditional market, as well as 5 supermarkets in 5 min walking distance.