As someone who's mostly self-taught (I have a BSc, but it's in health sciences), I followed a similar route to what they recommended. It gives you:

• income,

• experience working in software development - you will hopefully learn a lot from this,

• exposure to co-workers who you may be able to learn from, especially if your backend role is at a place doing ML.

With income, you can also afford to take more courses. Even if it's only the occasional weekend course, or something you work on a few nights a week, it can help you expand your skillset while gaining other practical skills (backend work with APIs, DBs, cloud infrastructure, etc are all useful).

After doing that awhile, you may be able to land a more focused ML role, or be able to do a master's program (which combined with your SWE experience will give you a leg up on landing the role you want). If you want to go straight into an ML role after SWE, you will definitely need project experience. But you can do that while working, if you're up for it.

One of the best ML people I know has a maths background, works in risk/finance, and is basically entirely self-taught. But the guy is brilliant and insanely passionate about what he does. I just mention him to show that you don't absolutely need to go the master's route. But it could be worthwhile when you can afford it, especially if you're lacking in maths.