Here's my 2 pence worth.
I'm sorry if it flies in the face of the others, but it is my own biased opinion based on 30 years in the engineering industry.
I started out with an apprenticeship in an aerospace company, and trod the Elec Eng path. At the time I much preferred Mech Eng, but felt the future was with electronics, (this was after the time electricity came in bottles). I am really glad I did. As people have said above, the big airliner stuff is feast or famine, and you are a small cog in a big wheel. It's my opinion that there are more jobs out there for elec eng, especially those with software experience. In the industry I'm now in, RF with software is the talent in demand. My belief when I started out is that an Elec Eng background would give me the freedom to work on stuff I was interested in, and possibly work in a small company where I would see the whole job, and my efforts could be seen, good or bad. I now work in a small company along side some really talented engineers, I also own that small company. I agree with the other posters that the qualifications are one part, experience is a bigger part. As an employer of engineers, I look at other stuff such as hobbies and pastimes, I'll hire the guy who fixes up cars and motorbikes with the EE over the guy with the PhD who has only been to school and finds the answers at the back of the book. I don't have a problem with PhD's but I've only met a few good hands on engineers with a doctorate. Most PhD's that I know now work in an industry not associated with their thesis, Chemists and Physicists now working as software engineers. In our company our in-house mechanical engineer and board designer is an aeronautical engineer, trained at Bae, but had to change career after a big lay off.
I have got to admit when you are starting out the grass is always greener somewhere else, In my early years I almost packed it all in to work as a sound engineer in a London studio. It is my point of view based on my experiences that Elec Eng would offer you more flexibility in the job market place, whether that was in Aerospace, Avionics,Cellular or the latest incarnation of WiFi, whatever. I also admit that some sexy electronics for aircraft are being produced by small dynamic companies, rather that the lumbering giants of 20 - 30 years ago. What ever you chose to do, give it your best shot, nobody owes you a break, you have to go out there and grab it with both hands.
I wish you very good luck in your future career, because that's gonna have to pay for the Cub.
Regards Pete