One should keep in mind that this is an article from a startup that wants to sell you billing software, so take some of the points with a grain of salt. Definitely some truths in there but having built inhouse billing solutions I can say that some of the issues pictured here are quite exaggerated. If you integrate with a payment provider like Stripe they e.g. already provide some support for tax codes, and even if not the rules aren't rocket science, every online store that sells internationally has faced the same issue.
In the end, if you let a third-party provider run your billing you'll be dependent on their solution, and if something is missing that they don't care to provide you're just out of luck, or back to building it yourself, which is why these third-party solutions aren't terribly popular. It's very difficult to build a standardized solution that just works, so relying on outside software can make your problems worse.