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I think this is partly semantics.

For example, consider designing a compensator for a power supply.  The transfer function under consideration is essentially the <i>open loop</i> impulse response of the system.  If that goes nuts, then you have other problems to fix first.  You are right in that it needs to be stable (not oscillate or grow exponentially or something) in response to a single blip.

But again, that's the <i>open loop</i> response.  You can certainly make a mess and cause closed loop instability with the wrong feedback, but that's not the transfer function being quantified.  In the end, of course, we do care about the closed loop transfer function.  By that time, we've designed the compensator (feedback) to make sure the system is stable.