I think it would be awesome to have these two modes to chose from.
Battlegrounds if my favourite PVP mode cause it's the most "plug and play" compared to IC/Cyrodil. Cyrodil is ofc the most fun when it works and you find people.
Battlegrounds is kinda weird cause it's Proc but no cp. I would love to try No-Proc Battlegrounds/CP Battlegrounds as the title says.