I’m hearing now that most chicken runs have a roof, is it really that important? They have a little roofed in area to the side is that enough?
Many runs have roofs, many don't. There are different reasons for that. A small run may not be that hard to roof, a big one presents more challenges. Some people build a roof to protect against flying or climbing predators. Wire can work for that.
Some people roof to keep rain out of the run to keep it drier. Wire does not work great for that. Even with a solid roof rain can blow in from the side also. A sloped solid roof can keep the run drier but they can still get wet in a rain, even with just a little wind.
I think you are mainly talking about keeping snow out. When my chickens wake up to a snow-covered world they are not going out in it for a few days. That is not necessarily a big deal if your coop can handle your number of chickens. But if your coop is kind of crowded it could become a problem. Think of the run as a pressure relief valve. If the overcrowding pressure gets too high inside the coop having a run available can relieve that pressure. If pressure does not build up it isn't needed.
There are different strategies to deal with this. Extra clutter in the coop can help, it can give them places to hide under, behind, or over. Over can be really beneficial, I often see my juveniles on the roosts with the adults on the coop floor when I go down to open the pop door and relieve the pressure that way on a normal no-snow day.
Some people are out there shoveling snow so they will come out. Others spread bedding or put pallets or lumber on top of the snow so they will go out. To me that sounds like work but it can be an effective strategy.
Another way is what I think you are after, keep snow out to start with. A lot of people do that, especially with smaller runs. That's not just a solid roof, snow can blow in from the side as well. I like that approach. You do that once, hopefully in good weather, and you are done. 21 Hens makes a good point. If your roof is solid it needs to be strong enough to hold up under snow load. But in an ice storm/freezing rain or certain types of snow a wire fence can pick up a big snow or ice load too. They also need good support. Some people go out and remove the snow to keep it from building up too heavy. I applaud them for having what it takes to do that but man, my bed is awfully warm on nights like that. At the same time, you do what you need to do.
It looks like you built that covered area as a dry place to feed outside. That's a pretty common solution. Will it provide a pressure relief valve for overcrowding if that is a problem? Will snow blow in from the sides. Can they eat without coming outside? Can they get over there if they need to, that may require shoveling or putting something on the snow so they will go over there. I don't know the answer to any of that. Hopefully your coop can handle your flock without any of this being an issue. I like easy solutions. But if you have issues maybe this will give you some ideas on how to manage them.
Good luck!