Like many online or tick box forms, which claim to be a Fire Risk Assessment, it seems to start from the point of view that the building is fine, and it is just a matter of managing the existing.
As has already been said, with many schools, this is not the case. After many years falling outside the legislative control of the Fire Service, where at best, Education Authorities were given a pile of "goodwill advice", for consideration and prioritizing, there is often a lot of work to be done achieve a reasonable standard.
I've seen no form based approach to Risk Assessment that incorporates professional judgment needed to benchmark etc etc.
That said, I do believe it is possible, using the guidance available, for a layperson to complete a suitable and sufficient risk assessment using a good template. Many Fire Services have very good ones on their websites. The more complex a building gets, the more difficult and time consuming the exercise becomes, at which point a judgment needs to be made between how much the laypersons time is costing, compared to enlisting professional assistance.
In terms of local authority schools addressing the issue of risk assessment on an individual basis, I believe this is a cop out by the Education Authority. The Responsible Person is the Director of Education (or equivalent). One Enforcement Notice served on a Director of Education has more effect than one served on every head teacher in the authority.