This will work perfectly fine in game, but there are a few caveats:
First off, most people like for their forts to look nice. Separate rooms helps that.
Secondly, it's easier for most people to remember exactly what is what and where it is if they're in sectioned off rooms of the fort.
Lastly, this affects the value of some rooms and that can be important. Especially for things like your dining hall. Allow me to explain: rooms that you designate from a piece of furniture have a value associated with the "room" that they're in. This room is just the area that you designate from that piece of furniture.
Say you're building a dining hall. When you set a table to be a dining room and specify the area around the table, that area is the dining hall. The bigger it is, the more valuable it is. Smoothing stone and engraving it makes it more valuable, and having more furniture in the area makes it more valuable. Having a valuable dining hall is a powerful way to keep dwarves happy. The same principles apply to the dwarves' bedrooms, and a few other things like offices.
The reason you wouldn't want to have such rooms randomly scattered throughout a 60x60 area is two-fold: you'll probably forget where they start and stop, and when you have rooms overlapping their value goes down (by how much, I have no idea). When you put a designated room in a physical room with walls, you can easily keep this from happening.
Overall though, it's certainly possible to run a fort with a couple of huge communal areas with everything scattered around in it. It's just less efficient.