Basic climate question

While I'm on the subject of climate, is there anyone with background in system modelling who can tell me what's wrong with the following. It  seems obvious to me, but I'm probably missing something important.

Take a global climate model, without human GHG emissions.

There should be no long-term climate trend, or at least none on the scale of what GHG emissions are supposed to cause.

Perturb the climate model in some way: maybe a major volcano, or a temporary solar variation, or something.

That perturbation, if it is big enough, will detectably affect the climate, including the global average temperature.

The perturbation is a one-off, so it will end. After the perturbation ends, does the global average temperature stay around its new value, or does it move back towards its old value?

My understanding is that the best current models of the climate exhibit positive feedback. That would mean that there are no forces that would cause the climate to move back to its old value. Block out the sun for a bit, the climate will get colder, and when the dust has settled (literally), it will still be colder than it was before.

For the temperature to return to "normal" after a perturbation, there would have to be net negative feedback: because the temperature is higher, something happens that removes the excess heat.

Therefore, without negative feedback, the climate would be a complete "random walk". It could be pushed up or down by solar changes, vulcanism, vegetation changes, even freak weather, and every such perturbation would affect the future climate forever.

That just doesn't seem remotely plausible to me. Surely such random events wouldn't balance so well as to keep the climate roughly stable for so long as paleoclimate data indicates?

I feel I must be missing something. Is there some way that the climate can exhibit positive feedback in response to anthropogenic CO2 emissions, as we are told it does, and yet recover from other random effects?