Nick,
Your assertion is borne out in practice. If mobile phones did affect aircraft systems then all mobile phones would require to be handed in on boarding any commercial aircraft. Alternatively, some kind of scanner would have been detected to scan aircraft cabins prior to take-off before every flight for phones still on.
I bet that most flights which take off have a mobile phone on in someone's luggage somewhere. If they interfered with systems even 1 in 1000 times then commercial aircraft would be falling out of the sky. They're not.
QDM