The word, "gemütlich" fills that role in German, although there is also a noun, "Gemütlichkeit."

There are plenty of cozy places in the US. If they get well known they are ruthlessly marketed as "Bob's Cozy Corner" or something similar. A chain of "Bob's" is opened where the coziness is Disney-ized and minimum wage worker's and sub-minimum wage waitstaff slave to extract every penny of coziness from Bob's.

Either the country moves on (e.g. Stuckey's) or Bob becomes a raging billionaire asshole fighting to oppress the people who both made him rich and made Bob's cozy in the first place (e.g. Howard Schultz).

There is too little "socialism" in US society for for most coziness to last a long time.