Scramble meaning "moving quickly and frantically" is much older, as in
When I saw the coffin I knew that
I was respited, for, as I judged, there was space between it and the wall
behind enough to contain my little carcass; and in a second I had put out
the candle, scrambled up the shelves, half-stunned my senses with dashing
my head against the roof, and squeezed my body betwixt wall and coffin. (J. Meade Falkner, Moonfleet, 1898).
Which quote I found via wiktionary, and then fact-checked against the Project Gutenberg copy of the book.
The etymonline link you link to gives this as being from the 1580s, (and predates it being specifically used of a way to cook eggs) and suggests it may be a corruption of scrabble.
It's not hard to see how a word meaning "move quickly and frantically" could gain a specific air force meaning among the RAF in the 1940s. The Battle of Britain started in July 1940, and the struggle for air-superiority over Britain continued until 1941 with the most intense fighting in the early stages (the British consider the battle as having been until the end of October 1940, the Germans until 1941 when air operations were focused on the Eastern Front, and Churchill made his speech about "The Few" in August 1940).
For this reason, The Few regularly had to scramble in the earlier sense with great frequency. Previous air battles where not as long-lasting, or had depending on aircraft that took longer to get into the air. That they called this scrambling and scramble developed a sense specifically relating to getting planes into the air is not surprising.
It's perhaps also worth considering that the procedures behind getting battle-ready planes into the air and ready to engage are different now as to then. There was likely more of a scramble in the old sense to a scramble in the new sense then than now.
There are two other words worth considering here.
Scarper is informal British English for "leave quickly", originally slang, and either from Italian scappare (escape), Cockney "Scapa Flow" (go), or a mixture of both.
Scram is similarly American English for "leave quickly", and also slang. It could itself come from scramble, or from German schramm. Some will tell you that it's an acronym of "Safety Cut Rope Axe Man" or "Safety Control Rod Axe Man" as it relates to nuclear reactors, but this is a bakcronym. This in itself points to a German origin, as while German-based slang was popular in the 1920s, it was less so in the 1940s for obvious reasons (the US Army slang FUBAR might be from furchtbar, with the well-known definition being a similar backronym for similar reasons).
I don't think that scarper or scram are the origin of scramble in this sense; I think it's from the earlier sense of scramble as I gave above, but I do think they could have influenced that rather than another choice (hurry, fly, get into that f'ing sky before Jerry blows us to bits) being the term that became common and developed an Air-Force specific meaning. This would presumably relate to the phonosemantics John Lawler mentions above.