we have no body and only flimsy circumstantial evidence ( some pointers she may have entered the river but no evidence she left the field yet) so far
That's not ' permissive policing' and remember that police do not make the laws but they have to adhere to it.
Worth having a look at the missing persons section on Websleuths. If you want the police to have the kinds of powers & the citizen to have fewer rights, like they have in dictatorships like Russia, then yes, you can have a situation where police can force entry to search everybody's home in that village with a fine tooth comb. ( As the law stands, they don't have those powers)
Know what you mean though, I think I'm becoming an old cynic, am flabbergasted that the media, some pundits and some of the public want instant results in 2+ weeks, as if they're watching a 6 part season on Netflix.
Were we always like that?
law on power of entry & search warrants
Powers of the police in England and Wales - Wikipedia