The two chips, one in the NES and the other in the cartridge, act as a pair. The "lock" in the NES looks for the "key" in the cart. Disable the lock, the door is always open. If the 10NES in the system does not detect the chip in the cart, it prevents the system from continuing to "boot". If the chip is disabled, then any converter should work.
I only have two "official" converters, both in a Gyromite and a Duck Hunt cart. I modified the one in the Gyromite to allow connecting Famicom games to it with the cases. Just requires removing about 1/8" off the ends of the fami side of the connector to allow the cart to slide over the connector. My dremel assisted me in that act.