I checked my updated edition of Swan's Practical English Usage, and Unit 494 now refers to something other than relative constructions. Unfortunately, relative constructions are not even listed in the index, under "relative" or "constructions. It's too bad, because I would like to have read the section to which CopperKettle refers, as his comments and answers are always useful and worthy of consideration.
That said, I tend to agree with the OP (struggling tutor) in that when you have the opportunity to be more specific, it is better to use "where" or "who" or "when" as appropriate rather than "which" or "that" for the sake of clarity and consistency. Perhaps I am being a bit old-fashioned about it, but "Isn't that the place where children under 12 can't enter?" sounds much better to my ear than "Isn't that the place which kids under 12 can't enter?".
Likewise, I also prefer "The Queen Alice is the place where I go the most" over "The Queen Alice is the place that I go to the most." Please note that I'm not claiming the original sentences are ungrammatical, only that I prefer the greater specificity and clarity of the alternatives.