Currently I am using Vector, which was part of the Mediawiki 1.35 bundle, but I have also tested other skins and the behaviour (missing search suggestions) is the same in any skin.
No errors in the developer console.
No errors in the Apache error log. (lots of PHP warnings that don't matter: deprecated hooks)
No errors when using the opensearch api. It works but I notice the opensearch api slightly differs from the search suggestions in the way it searches namespaces ...:
The documentation states I could supply the namespace within the URL, like "api.php?action=opensearch & search=N & namespace=12" *) This parameter seems to be ignored, the search result shows articles of the main namespace.
On the other hand I can supply the namespace as part of the search string "api.php?action=opensearch & search=Hilfe%3aN" *) The search results are then limited to the "Hilfe" namespace but shows all articles containing an "n", i.e. a fulltext search - in contrast to the search suggestions dropdown, where I expect to see all articles beginning with "n".
*)the editor rejects the full url (spam filter)