While a roll cage is never a bad idea (unless it's a road car*), there's nothing wrong with using cut slicks without one. If it was a likely to be a problem then circuits would require a cage, like they do when using slicks (which are much less progressive). If you're the sort of driver that is going to have an incident on a track day then the type of tyre you are using is not going to make a great deal of difference, in my opinion.
Combe's a bad example as there's a severe lack of run off at just about all of the corners, but even so, the kind of speed you are likely to be doing IF you did reach the tyres or barrier if say, you out brake yourself, would be no worse than if you had a mild road incident. You might argue that you're more likely to have a high speed crash on a track - yes, if you're racing. If you have a big crash on a track day, such that a cage would make a significant difference, then you were not driving appropriately for a non-competitive track day.
You can never rule out the driving standards of others, mechanical failure, or simple driver error, and for such reasons a cage is always a good idea for circuits - but I would definitely not see it as a requirement and wouldn't lose any sleep over not having one.
With that said, nice video! Your friend would benefit from some tuition but on the whole is doing quite well. He's a little early on a couple of apexes and needs to be smoother with the throttle (a few times he goes from brake to throttle, then a lift, then throttle again, while entering a corner - always a sure fire way to unsettle the car), and needs to sort out his grip on the steering wheel.
Any photos?
*Cages in road cars - bad idea, for two reasons. Firstly it creates a problem with the emergency services, should you be involved in an incident and you need to be cut free from the vehicle. Secondly, if you don't wear a helmet (which is illegal anyway in an enclosed vehicle) you are at risk of your head making contact with the cage in an incident.