On your first point. This is not a requirement by marketplace legislation/law. No compliance company is legally required, and most companies do their compliance work in-house.
I see, so get a random person (including the importer/seller) to understand, check, verify (any included) test report certification to ensure the batch being sold conforms to all safety and regulation standards.
If you're saying that's the case for kid's toys, then self certification in this category is an ass.
Self certifying (IMHO), children's product, as well as electrical products (unless very low voltage) is both reckless and can endanger lives.
An importer or seller of these products, will cut corners so they can sell their products. Most will not give a damn about consequences, nor have the correct insurance in place to cover themselves in the event of a claim.
I'll repeat, it's reckless and can be dangerous.
On point two. You are right that many certificates are rarely correct and that the Chinese do not (on the whole) take CE marking seriously. Do bear in mind that it is the responsibility of the "importer" to ensure products are compliant, as they will legally be responsible if they are the ones bringing product into the UK/EU.
The importer generally 'relies' on the certification it's given, without verifying if the tests, or indeed the certificates are genuine, relate to the same model/product or the batch they are being sold.
I can tell you categorically that the vast majority of 'China' testing labs, are (mostly) useless and have very little knowledge of the product, components, testing required and latest regulations. They simply copy what's online and offer limited pages of a full test report.
Unfortunately, most UK importers are attracted to the bottom cost price, rather than a quality product.
You know where that leads to . . . . .