Family can mean different things to different people. You can keep your eating extravaganza to immediate family or extend the invite to anyone your child considers family.
Again, who you choose to call family is not the important part of this minimalist birthday celebration. Food and focused attention on the birthday child is the point of the meal.
My kids both chose intimate friend parties this year. An intimate friend party is one with very few guests which allows for a more extensive experience.
A minimalist birthday party for kids can look pretty similar to a traditional birthday party. But instead of splurging on decorations you focus spending on the experience.