Gutenberg LinkControl suggestionsQuery not working

However, typing into the search input shows all post types (I want it
to just show pages).

That’s most likely because your code, as well as the example in the documentation, used the wrong property name — suggestionQuery — which should actually be suggestionsQuery (note the plural “suggestions”).

So try using suggestionsQuery?

<LinkControl
    suggestionsQuery={ {
        type: 'post',
        subtype: 'page',
    } }
/>