Is it short-sighted not to make eye tests compulsory for kids?

You need to be a little more informative. If lazy eye (amblyopia) is to be avoided it is important that vision testing is done at a sufficiently early age. This means as early as a useful examination can be done and school age onwards is usually too late. Many of us had school eye tests by the district nurse for no good reason.
Orthoptists are specially trained and skilled in examining vision in young children. In very young children the Health visitor often refers a child to the orthoptist with a query about the vision or a possible squint/lazy eye. The GP of course can also refer a child to the community orthoptist or to the local hospital eye clinic to see an orthoptist and ophthalmologist if necessary. Referral to hospital is not usually the first step and the eye clinics there are often horribly busy.
If a child is to have a sight test (refraction) by a local high street optometrist (previously called ophthalmic optician) I suggest checking BEFORE making an appointment that they intend to do a proper refraction (ie after cyclopentolate drops) as very misleading results can be obtained otherwise. Many optometrists are reluctant to do this and many are not comfortable refracting very young children.
Understanding the details of visual development and sight testing and amblyopia is not simple which is why we have orthoptists available to help with these queries. Orthoptists are not refraction experts and optomertrists do not have all the orthoptist skills though there is some overlap. If there is a problem the optometrist often needs the orthoptist to also check the child and if the orthoptist finds the vision subnormal the an accurate refraction after cyclopentolate drops to enlarge the pupil (and briefly stop the contraction of the ciliary muscle inside the eye) is always the next step.

Nice to see a subject properly explained. The extension to this must be surely at what rate would the commoner problems be found by a high street opticians. In any event surely the schools could be instructed that children have an eye test certificate from an optician before they can enroll. I accept it cannot be 100% effective but perhaps it is reasonably cost effective. Mr. Alexander might have a better idea if this screening would be helpful.

Which actually comes back to the missing information – approximately how many are currently referred and how many children have defective eyesight. Percentages – particularly with a big range fill me with distrust. Is it possible to see the raw data ?

I agree in principle that all kids should have eye tests starting at a suitable age. If, as Alexandrer says it is a specialists job, Then some sort of title should be given to these specialists and possibly train high street opticians in the skill.
This is out of my experience, but surely something can be done to protect kids eyesight and spot any abnormalities early on. Of course it should be free. Another problem is to catch ALL the kids!

Why can’t the Department of Health make it compulsary? What’s its excuse?