Rather than fiddling with the filtered_search JS, it's probably enough to not return issue weights in search results, and not to respect issue weights in the search query.
Following discussion with @DouweM, we're going to make burndown charts feature depend on both the burndown_charts and issuable_weights license feature. If we ever sell a plan that includes the former and not the latter, we can work on making burndown charts work without issue weights then.