Is it possible in practice to control the side effects of making changes in a huge legacy code base?

Maybe the software crashes when you write 42 in some field and you're able to tell it's due to a missing division-by-zero check deep down in the code base. Your gut tells you you should add the check but who knows if something relies on this bug somehow, plus you've never heard of anyone having issues with values other than 42.

At this point you decide to hard code the behavior you want for the value 42 specifically. It's nasty and it only makes the code base more complex, but at least you're not breaking anything.

Anyone has experience of this mindset of embracing the mess?

I believe this is called Microsoft Driven Development

(seriously though, this book has answers for you: Working Effectively with Legacy Code, by Michael Feathers)

All. The. Time. And I hate it. Imagine giving a customer a rebate based on buggy code. You fix a bug, the customer comes back and wants to check that the rebate was correct that last time. Now you have to somehow hard-code the rebate they did get so that your (slightly less buggy) code gives the same result. But hard-coding has the risk of introducing other errors on its own. Oh yes, and you've never enough time to do things properly because Customers (or maybe Management). A tangled mess of soul destroying lifeblood-sucking code and pressures ensues.

I've never seen code truly get that bad, but I can already think of several problems with that approach.

Do you really know all of the expected behavior you're hardcoding in? What happens if your hardcoded behavior is just incorrect enough that it breaks something somewhere else? How can you be sure that your test for that specific value is even correct?

I think the better approach is to let things break naturally and open a bug with your findings. You'd be surprised how often someone else knows exactly what's going on and can fix it correctly. Your hacks are not just pouring gasoline onto the fire, but opening a well directly underneath that will keep it burning for a long time.