As the team’s manager, it’s your job to get buy-in from the executives to gradually fix the mess. You don’t need to tell the team exactly how to fix it, but you gotta get buy-in for space to fix it.
One approach is just to say “every Friday goes to adding tests!” (And then when there’s some reasonable test coverage, make fridays go to refactoring that are easy with the new tests, and so on).
But this often fails because when Friday comes, something is on fire and management asks to please quickly squeeze this one thing in first.
The only other approach I know of is to get buy in for shipping every change slightly slower, and making the code touched by that change better. Eg they want to add feature X, ok add a test for adjacent existing functionality Y, then maybe make Y a little better, just so adding X will be easier, then build X, also with tests. Enthusiastically celebrate that not only X got shipped but Y also got made better.
If the team is change averse, it’s because they’re risk averse. Likely with good reason, ask for anecdotes to figure out where it comes from. They need to see that risk can be reduced and that execs can be reasonable.
You need the buy-in, both from the execs and the team. Things will go slightly slower in the beginning and it’s worth it. Only you can make sell this. The metaphor of “paying off technical debt” is useful here since interest is sky high and you want to bring it under control.