This is a tough tough situation. There are no easy answers or quick wins here. So before we even think about code, let's ask some questions...
1) You said you can't manage this team directly. Is it your responsibility to make this team successful? I know it's annoying to see a team with horrible code and who refuse to change. But is your manager expecting you personally to fix this? If not, just leave it.
2) Even if it's your responsibility, is this where you want to spend your time? As a leader you have limited time, energy and political capital. You need to decide strategically where to spend that time to have the best impact on your company and to achieve your personal career goals. The fact that you can't manage them directly makes me think that they're not your only job. If it's just one area of your responsibilities, I'd consider letting this team continue to fail and focus on other areas where you can make some wins.
3) Is how the business views this team wrong? They're making a lot of revenue with a very cheap team who seem to be very focussed on delivering results. Yes I know, it's annoying. They're doing everything wrong and their code is unimaginably dirty. But... They're making money, getting results and neither they nor the business see any problem. So again... should you just let it be?
4) Ok, so if you're absolutely committed that this code base has to be fixed... maybe you should just find a different job? Either in the same company or in a different company.
5) Ok, so it's your problem, you want to solve it and you're unwilling to leave. What do you do?
Well, anyone can make a list of ways to make the code better. Because this team has been doing everything perfectly wrong, it's not hard to find ways to improve: source control, automated testing, CI/CD, modern libraries, SOLID, clean architecture, etc, etc.
You can't quietly make the changes, because the team doesn't agree with you. And even if they did, this hot mess is way past the point of small fixes. You need to put in some solid work to fix it.
So you need buy in from management. You either need to deliver less while you improve the code base or spend more money on building a larger team. But since they see no problem, getting their buy in won't be easy.
Try to find allies, make a pitch, frame the problem in business terms so they understand. Focus on security risks and reputational risks. And don't give up. You may not convince them today, but if you make a pitch, they will remember in 6 months time, when this team is still floundering. They will remember that you were the person who had the answers. And then, they may come back and give you the time and resources you need to clean up the code base.
So in conclusion. If it's not your problem, ignore it. If you have other teams to manage that aren't a mess, focus on them and let this one fail. If you're going to be responsible for this pending disaster, quit. If you absolutely insist on making a change, start with getting buy in from management. Then incrementally work down the technical debt.