As a current councillor but also a CiLCA qualified clerk I did object to undertaking the qualification when it was first introduced but once I'd completed the portfolio, and it's not an easy one to pass by any stretch, I was glad I'd done so. No council is identical to another, we all have different circumstances, priorities and requirements and no course of study can cover absolutely every eventuality but it provides a good grounding in basic local government law and procedures and, importantly, guides clerks towards where to find relevant guidance for situations they may be required to handle on behalf of their councils. Significantly, it is the required qualification for obtaining General Power of Competence (alongside the electoral qualification and relevant declaration). So, in answer to the question can an unqualified clerk do the job, then yes of course they can, provided they have the experience and expertise but the council will have to operate without GPoC.
It is not correct to say that it is solely the remit of the SLCC or NALC to govern. The qualification is a nationally recognised one which is administered by the University of Gloucester (at least mine was!) and is subject to the checks and requirements of that institution. Whilst no system is entirely foolproof, education and training delivery bodies are very aware of opportunities for using AI, plagiarism and techniques that might be used by a candidate to submit something other than their own work. There are mechanisms in place to identify and deal with suspected cases.
There is currently no requirement for CPD (continuing professional development) for retention of the original qualification and there is always an argument that there should be. Members of the SLCC are encouraged to undertake regular CPD and training is offered by a number of different bodies (SLCC, NALC, principal authorities, internal auditors etc.) aimed specifically at Clerks although not all councils encourage or pay for their staff, or councillors for that matter, to complete training. That is a matter for councils to consider and budget for.
I cannot answer why your parish clerk has given incorrect advice, if that is the case, but there are always a range of reasons why that can happen in any role and is something that should be reviewed by the staffing committee.