Hi osca, and welcome to cruise-chat.
You're right in that Freestyle Dining is becoming more popular. A couple of other major lines, including Princess and HAL, have adopted some form of open seating dinner for some of their passengers. The difference is that those lines also continue to offer a traditional option.
But, back to your cruise...
I see from the NCL Web site that you have a busy itinerary: six ports in seven days, including Barcelona. A few of these (Rome, Florence, Nice) are a distance from the pier. That means that you will be spending a lot of time (and money) touring. So, you may very well spend little time in your cabin, but at the same time you might prefer having your own private deck/balcony on which to relax when pulling into/out of port. There's also the trade-off of whether you want to spend money on the shore excursions or the cabin. There's really no right answer -- it's a personal decision.
We tend to get balcony cabins as there are three of us, and on most lines the larger cabins tend to have balconies. I use them more as a floor-to-ceiling window as I am afraid of heights. But, when our son was smaller, we traveled in inside cabins and were fine.
I haven't been on Norwegian Gem (obviously
), yet her sister ship Norwegian Jewel is stunning, and there are many public areas from which to enjoy the view.