The EU-INC is EU designated without a sponsor, which is not permissible under EU law other than for the EU institutions themselves. This is one of the red lines for the design of the EU legally speaking
What do you mean by "sponsored by the member state"? Can you point to the exact regulation, treaty or directive that imposes this requirement?
The only exception to a legal entity are the EU entities themselves which are supranational.
This is what most people don't understand with the EU, that there is a very specific process to create law and limitations on what it can do. It isn't like the US and the EU isn't a country. I may appreciate you might be European but even within Europe the detail with it is where it matters.
But even if this were implemented using a Directive (note: the SE is created in a Regulation, that's the link in my previous comment), this still would not mean that the law or the directive would necessarily need to require a company to be "sponsored" by a specific state - whatever that means. (Again, it would help if you'd define "sponsor" with a specific legal reference.)
And even then, Von der Leyen could have been talking about changing the TFEU treaty itself, which is ambitious but certainly not impossible. But again, I see no reason why this can't be done in a regular fashion, just like the already extant SE Regulation.