We still debug and fix on production servers from time to time (once every few months), although we have intermediary staging environments and a full release procedure.

But, when sh*t hits the fan, the business enter in the "ain't nobody got time for that" mode and they urge for the fix.

That should never be necessary; if you invest in the appropriate automation you can get your regular deployment process down to a few minutes and then even emergency fixes can go through the normal channels. The benefit is of course mainly the ease of day to day deployment, but it does mean that in those emergency cases you don't need to worry about what you've forgotten to do, or how your changes might mess up the assumptions baked into the automated system.