Don't remember The Family. Look, lettuce and tomato can be OK on a Cuban, but you'd better eat it right away (or have a purveyor as thoughtful as Graziano here, who gives you a takeout container with the "salad" items and some extra oil & vinegar, so you can self-dress later). In Philly they offer lettuce and tomato, too, and call it a steak hoagie. But it's quite a different sandwich for it.
Cubans in Tampa and Miami are like tacos or beefs here (or steaks in Philly). They are everywhere, the locals seems to have a natural feel for what can and can't go, many are mediocre or bad, some are great, and on any given Sunday you might walk in and get a taste of heaven that is tough to replicate. In my view, it has the essential elements of a great American sandwich - great, very specific, local bread, and at least one meat that requires something more than a slicer and the Sysco truck. In the Cuban's case, that's roast pork.
Cf., Italian beef, Philly roast pork, NOLA debris poboy, etc.