Right (no pun intended). In general, a right to X means you can do X, not that you have to do X. But the original article said that the new law would add an inalienable right to remuneration, and said that meant publishers would be required to demand payment.

That made it sound like when a right is inalienable in the EU, you must exercise it.

In the US, inalienability affects how hard it is for others to stop you from exercising the right, both intentionally and unintentionally. It is still your choice whether or not to actually exercise it, and depending on the particular right you may also be able to freely agree by contract to not exercise it.