It depends how much you want to see and be in Rome. I personally love Rome but prefer other places in Italy to Rome. Rome is just another big city and I say this being a city girl and having seen a shitload of cities. The only 3 cities that I could spend more than a wk in include Paris, Tokyo and Manhattan. Vatican adds to Rome sights but otherwise, you have the top sights to see and that's a wrap.
I wouldn't do Italy or any European city for 7 days or less though. I would do 11 days min. European countries are laid out so that train travel works to see multiple sights. It's a different culture than US where you stick to one city at a time. Around the world, people have trains that make moving around easy
So if you only have 7 days, do Rome and take a couple day trips. Go back and plan a longer trip to hit Positano etc. if you can arrange a longer trip this time, hit Positano or consider Ponza or Ischia islands. Positano is truly amazing but you can also hit other towns not as crowded but be aware that Positano is the easiest to hit going that way because it's first up. Or go to Amalfi off season and hit multiple towns there.
Is Positano worthwhile? It really depends on what you like, how many days you have and what floats your boat. Going to Amalfi is about enjoying the water. It's not about beaches but sauntering and ambiance. It's about beauty that's definitive of a Mediterranean coast. It's just beautiful but it's about scenery. So if that floats your boat, yeah, go to Positano but yes you want an overnight stay there and a boat ride to get enough out of it.