You can have a singular rail network system managed by one public company that is exploited by multiple private transport companies. The problem currently is that in Europe the rails are often also owned by the companies that use the rails and they don’t want to share. And the different networks often aren’t even connected to each other. If you take the high speed rail from France to Spain you need to get off near the border to take a slower train to cross the border and then you have to walk to a different platform to board the Spanish high speed train. Since the Spanish railway company and the French one don’t like each other. Not to mention that not every country uses standard-gauge.

There is no real competition between the different railway companies in Europe. Almost every route is monopolized by the owners of the different rail networks. While I can choose between different airlines for most routes. Put the rails under a separate pan European public company and force the transport companies to share the rails would solve this problem.