First aid at work is first aid for work situations while mountain rescue casualty evacuation is just that.  We did have a qualified medical doctor on one site and he had to do the First aid at work course to become a first aider (he passed first time!)

Its up to you these days to decide on a pproraite first aid - HSE no longer approves training providers. But they do provide this advice.

https://www.hse.gov.uk/pubns/geis3.pdf

Its a bit of a dry read but it does say

Alternative qualifications – where such qualifications are being used for the purposes of the Health and Safety (First-Aid) Regulations 1981, the syllabus should contain appropriate content to address the findings of the first-aid needs assessment. This is likely to contain elements common to, or indeed all the topics listed in the syllabus for FAW (see Appendix 1). The duration of training (not including breaks) should be based on comparison of the proposed syllabus content with that for FAW and the time adjusted accordingly.

So might be worth getting the mountain rescue sylibus etc to guide your decision.

Its about the context.

Mountain rescue will be focused on the injuries and presentation associated with climbing / skiing / walking accidents and the longer hand over period getting the casualty down the mountain or airlifted to hospital.

At Work covers the whole (less fit) population suffering injuries common to the workplace and typically transferred to a hospital in under an hour.

Back when I held First Aid certificates I obtained Public but then had to undergo a similar course for At Work the difference being no paediatrics associated with the latter.

When I needed first aiders - I had some but not enough - I accepted different qualifications, mainly from the 1 day EFAW, until a course could be arranged. I did check what they'd done against the guidance - but as we had several sports injuries & minor bumps, I considered that somebody with a Sports or Mountain First Aid certificate could deal with it safely.

...there are a number of courses out there that enhance the FAW including those covering Chemicals and Confined spaces...the core syllabus is still FAW for the workplace and although you all may consider those that have other qualifications...FAW only exempts those with the qualifications as stated before...that said if all you need is a appointed person then it is covered with a little bit more....

Don't get hung up on moral duties here as we all have some part to play in attempting to save a life...look at compliance with FAW first and then you can enhance it based on your needs...