In QM its obvious that we need to normalise quantum states since their inner product squared represents a probability. This normalization leads to physical states in QM being represented by 'rays' of vectors in the Hilbert space (i.e. with an arbitrary overall phase)

However in QFT, the inner product of the states no longer has this probability meaning, so why do we need to normalise states (e.g. choosing to normalise $|p\rangle = \sqrt{2 E_p} a_p^\dagger |\Omega\rangle$).

What would break if we did not normalise the states? Also what would be wrong with a state $k | p \rangle$ representing a different physical state to $| p \rangle$?

Is it just so that the inner product $\langle q | p \rangle$ is well-defined, even if it no longer represents probability amplitude?