Currently, if an updateable synonym filter is included in a multiplexer filter,
it is not reloaded via the _reload_search_analyzers because the multiplexer
itself doesn't pass on the analysis mode of the filters it contains, so its not
recognized as "updateable" in itself. Instead we can check and merge the
AnalysisMode settings of all filters in the multiplexer and use the resulting
mode (e.g. search-time only) for the multiplexer itself, thus making any synonym
filters contained in it reloadable. This, of course, will also make the
analyzers using the multiplexer be usable at search-time only.
Closes#50554
b1b4282273