I have been burnt by this attitude as an employee many times. What people usually mean when they say "I want to hire someone responsible to own the project" is "I want to hire someone who will pick up my vision of what needs to be done, will make it happen, taking the blame in potential conflicts along the way and will then let me get the credit for it".

I ended up starting my own business and I think that's the only realistic scenario for someone who likes "taking projects from inception to delivery and who actively seek out projects to own".