Arianne 6 is not about competition, but about strategic independence at the first place.
That’s what the Starliner folks said. The fault in their argument was ignoring scale effects.
Ariane 6 buys Europe zero practical launch independence, other than maintaining the workforce (and accompanying skill set). If SpaceX blocks Europe, it’s game over for any constellation operator and, in all likelihood, the European commercial space sector.
Ariane 6 hopes to do 10 launches per year by 2030. That’s a few weeks’ Falcon 9s today. Each Ariane 6 launch requires subsidies to be competitive, and that’s assuming Arianespace’s forecasts hold. (They haven’t.) Every one of those euros could be used, instead, on R&D.
Ariane 6 uses cryogenic fuel. It has no landing system, mass-manufacturing site or refurbishment elements. That means that none of the foundational technologies for reusable launch are being worked on. (Ariane 5E would have been a strategic hedge. But Paris wouldn’t have it.)
The point is Ariane 6 enables nothing new. If SpaceX blocks Europe for some reason, with or without Ariane 6, Europe isn't going to have a LEO constellation. Keeping Ariane 5 (5ME [1]) and developing a reusable platform would have been a smarter use of resources. (It aso wasn't particularly daring, either, over the last decade.) Instead, the ESA gets stuck with SLS but for Brussels.
Falcon 9's fuel is cooled slightly, if not cryogenic (RP-1 becomes denser when cold), and of course Starship uses cryogenic methane. And they both use liquid oxygen, although that's an oxidizer, not a fuel.
An extreme illustration: would you say that Spain had strategic independent seafaring capability if it had maintained fleet a galleons to this day?
If USA will go hermit mode like at the end of 19th century, then they will have no other choice. And we certainly have signs of US wanting to go into isolation.
Vulcan's capacity for a year or so is already booked, and Ariane 6 is also fully booked for several years out. Others in the class are approaching first flight, but still lacking enough information to book a launch on.