Under Bernie a new tyre war didn't seem to be an option, unless you counted the backroom war to see which company would pay the most to be the sole supplier.
Now that he's out... I don't know. Pirelli deserve some flack for sure for creating tyres with such narrow operating windows, but at the same time, that's what the sport asked of them, and we're already complaining that it's going in the other direction. I can't see other companies wanting to take up the cost for that much potential bad PR, especially when no one knows what the future tyre requirements will be (wheel size, engine power etc).
Personally I think this whole problem has been amplified ever since they got rid of refuelling but needed a way to make cars pit because they're not designed to pass each other enough on track.