Survivor in 2000, American Idol in 2002, The Bachelor in 2002, The Amazing Race in 2001. That always felt to me like when they found their "reality game show" formula that was then replicated off of those base archetypes into the entire rest of the genre. So mid-aughts does feel about right for when the explosion happened.

The more pure reality shows like original Real World or COPS seem more like ancestors than anything and didn't spawn as much of an immediate copy cat proliferation. Real World if anything morphed to be more like those later incarnations.

Unscripted television had been bubbling beneath the surface for a good long while (game shows have existed since basically forever), but the 2007 writer's strikes was the catalyst that caused MTV-style trashier reality TV to take over.