I certainly wouldn't call it 'old fashioned' to be an employee, as society has been trending from small owner/operator business to large corporations since the industrial revolution... so it's rather the opposite
I think your question is more about settling- you don't feel like working for someone else instead of yourself would be settling, and think you could be very happy in the scenario you describe above. Many young men in their early twenties are all dreams and bravado, and are not very practical. Sure, a lot of your friends and fellow students might claim that the only job for them is to start their own fund, but check back in 20 years and see who managed to make that happen.
So follow your own heart, do what YOU want to do, and if that involves happily working for someone else your entire career the way your father has, so be it! Enjoy!
"Twenty years from now you will be more disappointed by the things that you didn't do than by the ones you did do. So throw off the bowlines. Sail away from the safe harbor. Catch the trade winds in your sails. Explore. Dream. Discover.” ~Mark Twain