That's all true, and it is inevitably the case given that, as you say, the undertaking of electrical work is not, itself, a regulated trade.Personally I'd disagree with that description. They are a trade group - calling them a regulatory body infers rights/privileges/authority that they don't have. They do have a regulatory function in as much as they require certain qualifications for membership and (from what I've read here) police standards of work by their members ... but that authority applies only to their own members and effectively the worst they can do is throw someone out of the club. ... It is this latter bit that means they've been given (along with others, like ESR who also seem to like talking about themselves as something they aren't) the right to allow their members to self-notify work. But that's different to them being a regulatory body as most people would consider it. ... Ie, it's not like gas work where, by law, you need to be registered with Gas Safe before you can legally work on gas systems* ...
However, if one focussses on just the self-notifcation, that is regulated (by law), so in that one respect the scheme operators are as close to being regulatory bodies as they could be. The only complication is that there is more than one of them, but an individual scheme operator can 'throw someone out of their club' - so, unless they can get into one of the other clubs (and hopefully they might talk to one another about people who get thrown out?) that means that they have exercised a law-based authoirty to prevent that person self-certfying. Apart from the complication of the multiple scheme operators, that is, IMO, a sort of regulation.
Kind Regards, John