I would argue Reality Shows are indeed a new form of low budget Game Show (someone is always trying to "win" something).
Reality competition shows are definitely a kind of game shows, but not all reality ahows are reality competition shows; MTV’s The Real World was not – nor was its loose inspiration, PBS’s An American Family. Expedition Robinson from 1997 (and the international version, Survivor, from 2000) really kicked off the reality competition format.