Wow, here we go!
The engine101 web site is great, and not misleading at all. nowhere does it infer that the jet exhaust pushes against anything. It is quite accurate and precise.
The balloon is just like a rocket, in that a stored energy source propells mass backwards, which reacts against the chamber to push the vehicle forward. A balloon stores the energy in its elastic walls, a rocket in the chemistry of its fuel.
A jet does suck in air and heats it to toss it out the back, so it is different in that respect. Were a jet to have to carry all its oxygen, it would have a vastly reduced payload. For a 767 with 100,000 pounds of fuel, if the typical 15 to 1 mixture ratio is met, it must carry 1.5 million pounds of air for the journey (or 300,000 pounds of pure oxygen). This would put a big dent in its payload, like all of it!