"when a source singer dies, does a revivalist singer with a similar repertoire then become a source singer to replace him/her?"

You don't have to wait for people to die - it's happening faster than that - and the terms "source singer" and "revivalist" are very dated and don't have to be rigorously applied each time a song is transmitted from one singer to another.

I've seen it happpen, in fact I've been indirectly involved. A song is put together by a "revival" singer from collected sources, recorded and sung live and that becomes the "source" for someone else to rewrite their own take on the song, and then someone else does their version of that song.

Those so-called "source" singers - where did they get their songs from? Some may have been passed on through the family, but a lot came from the radio, the music hall and the stage. They wouldn't have called themselves source singers, though they knew some songs were older than others. And as Steve Tilston never tires of telling us, ALL of those songs were once written by someone!