"But the LEA want them in school! It is a bit like expecting the vicar to send them to a humanist, the estate agent saying they aren't necessary!"
No, it is entirely different.
As a public body, the LA can have no opinion on the question of how people educate their children, so long as parents comply with the law. The situation in no way resembles a person who follows a particular faith wishing to promote it, or a businessperson who makes a living from commissions hoping to sell their services.
The LA is responsible for providing school-based education to those who want it. In my area, the LA also collects food waste from households for disposal. This is a public service which is OFFERED. I no more accept that a government body should push school as a preferred option, than I accept them telling me I must hand over all my food waste to them if I would rather compost it myself.
I would argue that providing accurate and complete information is part of their duty also. They certainly should never misrepresent the law, as the vast majority of LAs do.
It's rather strange to say that people should have the sense to google something if they do not know it exists. By the way, the .gov.uk website's predecessor (I forget what it used to be) had very misleading info on home education. It has taken years of pressure from home education organisations and individuals to get them to correct basic factual inaccuracies.
I do agree with you that it is inadvisable to ask LAs for information about home education. But this is not because they should not be expected to provide correct information. It is only because they have such a shockingly bad track record on actually doing so.