a) to have passed the driving test or be supervised by someone who has, b) have valid insurance and c) be driving a car under 3 years old or one with a valid MOT certificate. Why? To minimise risks to other people
Legislation governing road usage tends to focus on the groups most likely to cause damage either because the group itself is numerous and/ or because the form of transport is dangerous. Motor vehicles have high kinetic energy and there are many of them, which is why they are the focus of most legislation.
Horses are less commonly found on the roads which doesn't mean they are necessarily less of a risk when you do encounter them, but does mean you're less likely to encounter them in the first place. Also, they usually have lower kinetic energy. Yes, they can weigh around 500kg and can travel around 30mph but that's still significantly less energy than a car doing 50mph.
In terms of testing, I could encounter a driver who passed their test 40 years ago and is assumed to be competent because they've continued driving and haven't been caught doing something that warrants retraining. It's better than not having a test at all but does not eliminate risk. There is a riding and road safety test for riders which is voluntary.
In terms of insurance, anyone on a riding school horse will be insured as riding schools are legally compelled to have insurance. The vast majority of private riders will have cover either through an insurance policy or through membership of competitive bodies such as British Dressage (it gives you some idea of the possible risks involved that BD insure me for £20 million of damage on a membership of about £70 a year). it would be interesting to know if compulsion would increase the % of insured riders. My bet would be that there are more uninsured drivers on the road than riders full stop.
It doesn't make sense to enforce an MOT on horses. 5 stage vettings are valid on the day they're done and no longer than that. I could prove to you that I can stop my horse but that would be covered if you thought riders should be licensed, rather than assessing the horse per se.
Similar debates go on with cyclists. The long and the short of it is that the effort required to enforce these things is not worth it for the lessening of risk that might result. Most adult riders have a driving licence anyway. They know about roads and the Highway Code. In fact IME vulnerable road users know the HC far better than people who drive but don't cycle or ride horses. They're forced to in order to reduce risks to themselves.