I’ve had so much trouble with tooling

Obvious skill issue. /s

I don’t feel like I can make any impact

Assuming you don't just want to vent, but maybe get some sort of unfounded and useless opinion from a total stranger to go along with your misery, let me oblige. I've never been a cog in a truly large company, but I've been rubbing shoulders with managers and C-level just barely enough to begin seeing "their side", and consider it in my interaction with cOwOrkers and management to effectively act as a conduit or translation layer between camps. It's a metagame you'll have to want to play, but it can be rewarding to finnagle the systems of human psyche to have a positive impact, not only for yourself.

From a manager's perspective, everything has a cost. It's called tech debt for a reason. As long as paying the interest on your tech debt outweighs the perceived cost of paying off the debt, you're doing the right thing from a business perspective. In addition, be keenly aware of the fact that you personally will impossibly know the huge dark-company iceberg that is the existing tooling and its context. I guarantee you that there's so much unknown to you, which makes the proposition "I would fix this in a few days, just let me!" seem rightfully preposterous, and thus actively dangerous, as you obviously don't fathom the full complexity of the larger issue and would necessarily break things for others, despite your best intentions and competence.

Yes, the tooling certainly is utter garbage, and they should never have locked themselves in such a horribly broken situation, but there they are, and you can't just "fix this" without understanding the implications. Many devs, myself certainly included, easily fall into the trap of seeing an obvious and clear way of fixing everything™ in a matter of minutes, and realizing too late that "just one more patch" is an endless pursuit, all while breaking user-space for others on the way. It's a meme old as dirt among hackers.

Assume you don't know all the details, and really nobody does anymore, and you'll live a happier life if you leave this (business) decision of letting code rot to others. It's their money they're wasting.

What you can do, however, is gold-wrapping the pile of shit you need to work with. Find creative ways to wrap byzantine processes into a bunch of scripts for yourself to automate and generalize your tasks specifically, and watch your productivity soar and your mental health rise, and offer that as a working band-aid to your peers and manager. That's not only a better use of your time than seething over how shitty everything is, but also plays very well into a devops mindset and is actually kind of fun. It also comes across way better than being the "new guy who knows everything better, but actually hasn't seen nothing, yet", which leads to people not listening you in the first place. That vibe is an instant turn-off for any lead, and leads to shadow-banning you from interesting, fruitful conversations you seem to look for.

Obviously I'm confabulating all of this, and this may very well not apply to you or your situation at all. This is just a behavioral pattern I've seen way too often, and it's helpful to detect this in oneself to prevent unnecessary frustration all around.

My plan

is possibly your best option. Go ahead. You have no obligations or "loyalty" towards an uncaring entity holding you back.