Hmm. We don’t really have quite the same heated debate about “amateur” vs “professional” in UK equestrian sport. Amateurs (who make their main living at something other than horses) competing against professionals (who make their main living riding and training horses) is to be found in most/all disciplines, even at the highest level. True amateurs run round 5* Badminton, there are even amateurs in top horse races (though usually National Hunt). With the exception of racing, there really isn’t enough money available in horse sports to support many professionals. Professional Event riders are a relatively recent phenomenon, since only the 1990s or so. Curiously, the most heated argument I can recently recall was professional Event riders running in unaffiliated competition where there were some good prizes and the amateurs were really up in arms about it! In BE Events it is simply normal that some riders are pro and the most amateur - but it is the horse that moves up the levels rather than the rider. The one discipline where “pro v am” status is regularly debated is Showing where presentation and ring craft is vital and the pro riders probably have the edge but amateurs frequently do still win. It is also the most subjective equestrian discipline.
To answer your question about qualifications and teaching. The BHS is constantly evolving and updating it’s qualifications. Currently, the system runs from Stage 1, basic horsemanship and riding skills easily achieved by a large number of people, up to Stage 6 which is the BHS Fellowship and extremely difficult to achieve. It is possible to focus on riding, or horse care or coaching as one progresses upwards. The language is no longer about “teaching”. The old Assistant Instructer (BHSAI) is Stage One and many people take it out of personal interest, such as they own a horse, they want to own a horse, even to fill the gap between finishing school and going on to university, or they take it as the first recognised step in an equestrian career.
It is not compulsory or a legal requirement to have BHS qualifications (or Association of British Riding Schools qualifications, also a system of education) to teach riding and riders. Anyone can offer lessons. BUT. Riding Schools must have a licence from the Local Authority, issued after an annual inspection, and adequately qualified staff helps in that process. BUT insurance premiums for Riding Schools are pretty brutal and having professionally qualified staff usually makes them cheaper. Accredited Trainers and Coaches can obtain professional insurence through the BHS which is very helpful. Most beginners are advised to look for BHS accreditation when searching for a place to learn to ride because they can be certain of consistent standards as a minimum. There certainly are unqualified people who teach on their own horse - who undercut accredited riding stables giving lessons - but when there are so many better qualified people who are properly trained to teach and to coach then sensible people use them. The specialist accredited coaches in Show Jumping, Eventing and Dressage generally build on the basis of their BHS qualifications.
Not a perfect system by any means. The BHS is a charity and has no Government funding.