The graph builder functionality was introduced in 2016 by Ice2cda02f597c6a90dfccf8b1a14e153ae624a5a and has seen only one minor change since that year.
It appears that polestar (the main graph builder component) is deprecated and so is its successor: T291928
The graph builder functionality will not work on any wikibase query service other than query.wikidata.org due to the hardcoded "whitelist" and GRAPH_QUERY_PREFIX (see: T357115 for Wikibase cloud specific issue). Given the https://www.wikidata.org/wiki/Wikidata:SPARQL_query_service/WDQS_graph_split the graph-builder functionality will be impacted already on https://query-main.wikidata.org/ and https://query-scholarly.wikidata.org/ since graph builder is making requests to the different dataset on query.wikidata.org
Before we consider fixing these issues we should decide if the functionality is worth keeping and is used (on wikidata; obviously it's not used on any other wikibase where it is broken by default)