You are right. Product Filter doesn't work with search result page. We tried to implement such feature as part of the ProductFilter addon, but it didn't work fast and seamless. The reason is the same as for search: we need a more professional tool to search and filter products.
That's why we extended CloudSearch and release CloudFilters, cloud base engine on Solr. Also, CloudFilters has a unique feature. It uses machine learning to understand what kinds of filters need to be shown for entered keyword.
For example, if you sell apparel and a buyer enter "blue jeans" CloudFilters will show only the filters connected with trousers, but not show filters of t-shirts: