You get to vote directly for the people with the power to invest and dismiss the Commission, just like you can vote for MPs with the same powers towards the Cabinet. Being a member of the House of Commons is not a requirement for being appointed to the Cabinet, so you haven't necessarily voted for the Cabinet member either. (Non-MPs can, and have recently been, named peers to join the Cabinet.) In other countries being a member of the national assembly is not a requirement to join the Government, so you can't expect the entire EU to follow something that may only be done in the UK.
Finally, commissioners are appointed by their respective governments (who answer to their respective parliaments). If you want the commissioners to be drawn from the EU Parliament, EU would have to change from being a cooperation between sovereign European states primarily represented by their governments (as is usual in international affairs) to a European federated state where final power rests in the federal EU Parliament. That may be viewed as more democratic (I don't agree, though), but I think few Europeans want to go that far now.