Much easier to make a radio antenna than a whole airplane.
The film indicated that the radio antenna was ripped off and that is why they couldn't just call for help. An antenna is just basically a stiff piece of wire cut to the correct length, and even this is not super critical. Not only this all commercial aircraft are required to monitor the emergency radio frequencies while in flight. Also at the time of this movie there was been a satellite constellation that listened for broadcasts on 121.5 MHZ a frequency that every aircraft has by international law.
Instead of spending all that time working on building a airplane why didn't they just make an antenna and call for help on the emergency channel.
Okay so maybe since the aircraft started with Coffman starters here was no working battery on board. Still it would have been easier to remove a generator from one of the engines and build a hand crank setup for generate electricity for the radio.
Guess that would not have been as exciting a movie. I can give a pass to the 1965 version because of the lack of satellite and possibility of over flights by a aircraft likely to hear their signal. But in 2004 this made no sense.
Any sufficiently advanced technology is indistinguishable from magic. Arthur C. Clarke