That said, I do think there are two standout foods that are ubiquitous in America although they vary by region. 1) bbq; 2) the sandwich. Behold the great American sandwich in all its varied forms:
https://www.nytimes.com/interactive/2015/04/14/dining/field-...
I think BBQ will eventually get it's due. We're seeing a lot of chefs take common, cheaper dishes and make them more upscale and interesting to larger audiences and foodies. One of the best dishes I ever had was dinuguan [1] at Paul Qui's (Top Chef winner) restaurant in Austin.
I don't think Americans have a huge problem with food as the parent suggests. I'm biased of course but I prefer it how it is. We don't need a singular identity. We have places like Napa, Vegas, NY and San Fran that stack up against any city in the world, even though our history is short. We attract top talent from everywhere and the melting pot delivers vastly different experiences. You can come to the US and visit New York, New Orleans and the Pacific Northwest and get amazing food with completely different styles and cultural identities.
Why do we need a unified food? Why is that important?
I also like to roast potatoes in it, as well as cook mac-n-cheese (in a shallow dish) to get that smoky flavor.
I'll never go back to a regular bbq or grill - even gas. The electronic controls of the Traeger (and I imagine other brands) make things dead simple; set the temp, throw on the meat, and let it go - return back after however long the meat needs. Don't peek under the lid. Remove, let rest, then enjoy. Best and easiest BBQ ever (I always hated having to babysit my bbq, checking temps, adding wood, making sure the water bath was right, etc).
I do use a small propane grill during the week sometimes though (small Weber Q which I recommend for this purpose). I've taken it to the beach and to friends before.
Def shouldn't be peeking as you mentioned. As the saying goes - if you're lookin you ain't cookin!