I don't think you're using it as intended. I also want to use it the way you do, but I think most people don't mind missing tweets. They're just dipping their toes in the stream. Facebook works the same way. I get FOMO.
I think it really depends on how many people you follow. Less than 50? You wouldn't want to miss anything. More than 500? There's no way you could read them all. As I followed more people I transitioned from FOMO into 'dip my toes' mentality, and I still enjoy it, and much prefer it to Facebook's over curation. I like that every time I check Twitter I see new things, which isn't the case on FB.