Hiya Anneliese!
Welcome to planDisney!
My family loves Italian food. It's cheesy, carb-based, and delicious. What's not to love? I wish I had a chance to experience the Pizzafari Family-Style Dining but I think I missed my chance. You are correct in that it is not a bookable experience as it is temporarily available.
Pizzafari has reverted to a Quick Service dining experience only at this time for Lunch and Dinner. There is no information about when it will be returned.
If you are looking for a similar menu then I would consider checking out
Mama Melrose's Ristorante Italiano in Disney's Hollywood Studios which has Vine-ripened Tomatoes and Fresh Mozzarella, Mama's Caesar Salad, kid's pizza, various pastas, and a Mini Cannoli Trio. I had a great time eating here on my last trip with my family. While it was not family style we did find the dishes to be hearty enough to be shared easily and yummy. My favorite Italian restaurant is
Via Napoli Ristorante e Pizzeria in EPCOT which has a delicious Mozzarella Caprese, Cesare (Caesar Salad), and one of the best pizzas in Walt Disney World. As a New Yorker, who proudly considers this her home state to be the standard for amazing pizza, I enjoyed the pizza at Via Napoli Ristorante e Pizzeria to be amazing.
If you are looking for family-style feasts then consider
Garden Grill Restaurant in EPCOT,
Roundup Rodeo BBQ in Disney's Hollywood Studios,
'Ohana at Disney's Polynesian Village Resort,
Sebastian's Bistro at Disney's Caribbean Beach Resort,
Liberty Tree Tavern at Magic Kingdom Park, and
Whispering Canyon Cafe at Disney's Wilderness Lodge. None of the ones I listed are Italian fare but there are quite a few delicious options that will please anyone! My personal favorites are 'Ohana and Liberty Tree Tavern. Both have a delicious bread pudding dessert that makes you regret having eaten so many delicious foods right before.
I hope this helps and if you have any more questions please feel free to
let me know!
Happy Planning!
-Nimmy