the least bad case is that you realize why nobody wanted to go ahead with your proposal in the first place but you caused no harm other than wasting several months of your own effort.
the worst case obviously is that you caused a huge amount of problems for the firm and people are aware of where those problems came from.
But yeah, I've done it.
For example, is there some process improvement, monitoring tool or logging solution that would reduce firefighting? Something that would immediately help you solve the next issue more easily? Do it while working the issue? Can you rewrite the tickets so they are no longer poorly planned, making them easier and quicker to implement? Do it while working on the issue.
When work is not optimal due to time pressure, there are probably costs related with that. Saving time and resources requires longer-term perspective and making some investment. But often there are things that require only smallish investment, that you can do.
When you save time or resources for your company, re-invest those into some little bit bigger improvements. If you take flexibility and get good results, you will probably be granted more flexibility.
I do this so regularly it is kind of a second nature to me. Sometimes is backfires, but if you really know what you are doing, mostly you will be fine. Might even get rewarded or promoted.
The secret is - and this is wielded a million times a day in a million small ways - of course we do. Lots of people do in lots of jobs, whether intended or not.
You just do the thing. You don't ask about doing the thing, you don't put in a project planning proposal, you don't beg your manager to devote 20% of your time to the thing, you just go and do it because you probably can. And as long as you do it, and it goes well, and nobody really notices until it's working, then it turns out you did have executive powers the whole time.
Just do the thing. Ask forgiveness, not permission.