Peter et al:
The old PE mags I have show a couple of solid state projects for doing what you want, but the articles are at least 20 years old so that means the PC board would not be available from the listed source, although the parts are all still common. The sequential flasher on the Goldmine site is the same thing, so I recommend you use it to avoid making a circuit board and hunting around for the other parts.
If you intend to substitute the LEDs with others more suited to your runway lights, let me know and I can search my catalogues for suitable parts for you.
Your motor and reed switch method will work, but it will not be as reliable as the electronic version and the motor will chew up batteries and spit them out compared to the electronic type. The motor type will also be bulkier, plus you have to putz with modifying the radio servo and mounting the reed switches, so for all those reasons, if it were my project I'd go with the electronics.
Let me know if I can help at all and good luck,
Bruce