Having trained and volunteered with CAB for a while, I think each case would be taken on its merits.
They would undertake to enter into dialogue on behalf of a client if it was clear he or she lacked the skills to attempt this themselves - e.g. illiterate/learning difficulties.
However, as you mention, the advisors are pressed for time, so they are probably NOT going to take on letters and phone calls for someone who can clearly express themselves as well as the OP has here. They will give hints and tips about how to approach the dialogue - what to include, what not to include etc, but it's all about helping clients to help themselves, NOT taking over from people who have the wherewithal to do it themselves.
If the OP's own efforts failed, and he was clearly out of his depth, they might intervene to try to resolve things, but I think that would be last resort, not first resort.
He's clearly pretty competent to express himself, so I think they'll want some indication he's at least tried, before wading in.
Although many CAB volunteers have some sort of legal background (in my intake, about half did), they are nevertheless lay people, NOT authorised, regulated or insured to act in legal proceedings. They can advise clients about the practicalities of how to make a claim in the Small Claims Court (which does not usually require a solicitor), OR furnish a list of local solicitors for more complex or higher value cases.
The CAB does retain a few paid solicitors, but not very many - and these would mostly be for potentially landmark cases affecting a lot of people. If you go in off the street, with a minor consumer dispute, you're very unlikely to get the services of a qualified solicitor, and the advisor you get may or may not have any legal qualifications at all - it's not a prerequisite for working there - but they will know basic law about "common" situations.