It's not just F1, but open wheel racing in general. Mikhail Aleshin was almost suffocated under Juan Montoya's car in Toronto, and I'm tempted to think of his Fontana crash as well as Dario Fanchitti's last year, but I don't think an enclosed cockpit could have helped either.
But let's get down to business, this is going to be a long post...
Let's think about the cons: Obstruction of view, lack of air flow and the ability to get out quickly.
Watch the video above, with the model tested, there are no support beams that would obstruct the view, but there is still the issue of dirt and debris. The first thing that comes to mind is tear offs which are currently used on the driver's helmets quite obviously. All top three divisions of NASCAR have tear offs on the windshield that can be torn off during the pit stops, so do the same thing. So there are answers here.
Now onto the matter of airflow, and I am going to reference NASCAR again. Obviously, NASCAR series have engines in the front, so airflow is almost even more of issue for them. I know they have a special circulation device that circulates the air within the confines of the helmet and keeps it cool, so I am sure F1 could do the same. I am also think that F1 could figure out a way to circulate and draw air from outside the car so if a fire were to be started, the driver would still be safe from the smoke.
About getting out quickly. The same was said about multiple point harnesses, that they would trap the driver and keep them in the car. I think with enough time put into development, F1 will figure out a way to eject the cockpit quickly allowing them to get out.
Another issue I would like to discuss is that a cockpit possibly could not have prevented the Bianchi incident. As seen in the video, the cockpit would deflect the force of the impact. Where would it go here? I am no physicist, so I don't have the answer, but I know it must go somewhere.
Considering what I said above, I do not see why this event specifically brought this debate up. This all could have been prevented if speeds were enforced in the sector and more so if the safety car stayed out until all safety vehicles got behind the wall.
Last, but definitely not least, is if this does happen, it is not the end of open wheel racing. Let's go back in history before side pods and wings were introduced to open wheel racing. Wings are not that good of example, but point is things change with time. A better example is the side pods. I am way way too young to know what people were thinking back when they were introduced, but I have a pretty good idea. "The space between the wheels is no longer open- we can't call these open wheel cars any more!." In my honest opinion, open wheel cars are open wheeled because they do not have body parts covering the wheels such as fenders, so the spirit of open wheel still exists, it has only evolved.
We are on witnessing change, we are witnessing the evolution of open wheel racing.