The problem with this plan is corporate politics. Say that OP takes on this challenge. He makes a plan and carefully and patiently executes it. Say that in six months he's already fixed 30% of the problem, and by doing do he meaningfully improved the team's productivity.
The executives are happy. The distaster was averted, and now they can ask for more features and get them more quickly, which they do.
Congratulations, OP. You are now the team lead of a mediocre software project. You want to continue fixing the code beyond the 30%? Management will be happy for you to take it as a personal project. After all, you probably don't have anything to do on the weekend anyway.
You could stand strong and refuse to improve the infrastructure until the company explicitly prioritizes it. But then why would that job be better than just taking a random position in a FAANG company? The code quality will be better and so will the pay.