If you are not doing notifiable works (as defined in Approved Document P of the Building Regulations), then you don't need to be registered with a scheme.
However, if you're doing domestic, then it won't be long before you start needing to do notifiable work (mainly new circuits, changes in special locations, new consumer units - if you're in England, in Wales the list is slightly longer) and so it's beneficial to be part of a scheme as you can just do the work and notify it after the event.
If you're not part of a scheme, you have to apply to the local building control department to see if they'll accept your quals as proving your competent... if they do, you'll still have to pay them quite a bit for notifying work and you still have to do it in advance of carrying out the work. If you're not deemed competent by them you end up paying planning application fees I believe. Suffice to say, it's much easier being in a scheme. I carry out the work, issue the relevant paperwork to the client and if required, submit a building control notification. I charge the client the for the latter, but it's a token admin fee instead of hundreds of pounds.
For everything else you just need to be competent.... competent to install, inspect, test and document according to BS 7671. Passing exams is one way of demonstrating your competence. Using your example of two guys (one qualified and one not), if they both install to BS 7671 and nothing goes wrong, great. If something goes wrong however, the guy with the qualifications may be treated with a bit more leniency that the unqualified guy because it could be argued that Mr. Unqual was acting in a deliberately negligent way by undertaking work he wasn't qualified to whereas Mr. Qual could have just made a mistake. I'm not a lawyer, but that's my take on it.
Trying to shortcut learning for this job isn't good, and some will say I'm a fine one to be talking because I went the short course route, but that was backed up with lots of relevant experience and an understanding on my part of my limitations (which I work within), but I am also still learning and having completed my level 3 exams and AM2. It's a career in which you will constantly need to learn. Once my NVQ is out the way I'm looking at EV charge points, fire alarms and emergency lighting so I can expand the range of services I offer my clients. I could offer them now but it would be a reckless thing to do as I know I'm not competent to do so.