I don’t have any experience of the F1 and shotgun mic, but looking at the spec, the mic sensitivity is −39 dB/1 Pa at 1 kHz, which I think is 11.22 mV/Pa. The ME66 is 50mV/Pa, so a lot more sensitive.
Having a very sensitive mic is not the be all and end all. The sensitivity of the ME66 is probably the reason for the issues with cable noise. A sensitive mic will also pickup more unwanted environmental noise - the defraction tube of the mic filters out some unwanted noise, but it stands to reason that using similar defraction tubes, the more sensitive the mic, the more unwanted noise you will get.
I note that the F1 comes with a hot shoe suspension (or perhaps it is an accessory). Anyway, unless you are mounting the recorder on a camera, you would need to find a way of mounting the device on a pistol grip (perhaps something from SmallRig?). I think if you directly held the recorder, as the recorder and mic are attached, handing noise may be an issue.
I suspect that the F1 is really aimed at vloggers etc - A recorder that can be easily camera mounted, has moderate sensitivity and can accept a high maximum SPL - ideal for close action in front of the lens.
I am sure you could get good recording of birds, but I am not sure this is the kit if you want to record birds a medium+ distance.
Cheers
Jon Bryant