I'm not sure what you mean by no bitcoin. Do you mean it would morph into something else or cease to exist?
That's a discussion that broaches a lot of people. You'll only read opinions since there is no one who can define what is Bitcoin. I personally define "Bitcoin" a computer program that is compatible with those consensus rules. Any other program that defies them should not be called "Bitcoin".
Some Bitcoiner refer
https://en.bitcoin.it/wiki/Principles_of_Bitcoin when talking about what define "Bitcoin".
At some point in time (I'm not sure when since I don't know if doubling the transaction rate will double transaction fees) people will look for a cryptocurrency with lower transaction fees.
I think you meant "doubling the transaction rate will halve transaction fees".