The Embedded Search is currently only used via the central search using "All Objects" or the "Free Text" attribute under a specific Business object: 
This search always search for all the indexed attributes. But when you have to search for a specific attribute via the central or the advanced search then the search is executed to the database. If the TREX powered Embedded Search would be used instead, then the performance could be increased dramatically. The next screen shot shows the standard setting of the advanced search for Accounts in the Business Role SALESPRO:
As you can see in the next screen shot taken form the test report ESH_TEST_SEARCH it is technically possible to i.e. search for the CITY attribute using the Embedded Search. Even a "Fuzzy Search" is possible.
My wish is that the advanced search would detect that the embedded search is active and search all attributes which are indexed using this search option instead of the search using the database. Especially in the case of the Business Partner that would have the advantage that the fields NAME1 and NAME2 which are limited in the database based search to the first 25 characters would be search for all 40 characters.