The EU Parliament was given a vote on von der Leyen. It was a ballot with a single name on it: hers. By the time it go to the rubberstamping stage the decision was already made. The MEPs couldn't propose other candidates for the vote, and the Parliament can't propose changes to the law either, so it's not the legislative branch of the EU. The legislative branch of the EU would be... the EU Commission. Which is also its executive.
The way von der Leyen was selected is a secret. Nobody knows how it happened. She didn't run in a democratic process of any kind, so she isn't a politician. If you ask EU fans they'll tell you the heads of state selected her. We have no evidence of this. That's the written process, but no records were produced of any such meeting, or a vote, or however it is that this decision was theoretically made. She could have been presented as a fait accompli by a single country, other countries could have been bribed, they could have been excluded entirely. We'd never know.