Much of that is at least for my company handled by our accounting company. We just print the correct VAT on the invoice, and report the same VAT to the accountant and they take care of the rest. The shop/payment processor etc doesn't need to be integrated to any of it. Though I have to post-process Stripe's reports, as they refuse to include the used VAT rate in there, despite them knowing it. Stripe does try to sell the tax service to us, but I refuse.

You can simplify for your use case (only B2C or you refund VAT afterwards for B2B, you only ship from one location, custom invoicing), but that’s what it takes to implement it correctly on platform level.