I think you make it seem like EU doesn't care at all about what member states do in regards to taxation but there's many limitations to what can be done by any member state in order to harmonize and prevent corruption etc. This in practice makes the EU have a lot of say in regards to taxation. Moreover the EU has special rules to limit moving funds to jurisdictions that have taxes that are deemed too low (read tax havens) - this directly implies no member state has agency to lower their own taxes as much.
Here's some example limitations: https://eur-lex.europa.eu/EN/legal-content/summary/tackling-...
I focused on direct taxation, but in indirect taxation I think there's even more examples.