I'm not so sure we really understand them. One the one hand, zeros are more fundamental than poles - poles can be reassigned by feedback, while zeros cannot be moved, they stay right where they are. Yes, stable zeros can be canceled - but they don't really go away, only become hard to observe - or unobservable if you believe in fairies and exact pole-zero cancelations.
On the other hand the very existence of a zero is somewhat arbitrary. A pure delay has no transmission zeros - what comes in gets out, at any input frequency. But a finite-dimensional approximation to a delay has transmission zeros. How can something apparently so immovable appear only in approximations?
The question is of course about the definition of the zeros in terms of the most fundamental description of a linear system, as a convolution of a kernel of integration with an input signal. Definitions in terms of transfer functions or state-space realizations are available and coincide.
Perhaps the definition of a zero as a transmission blocker is not so fundamental. I don't know, and I'm not terribly confident anyone does. Am I wrong?