The need to be switched manually could be the reason for the 70% discount, but it might noit be a bad idea, and I don’t think you need an Arduino to fix it. If you propose to do anything, you are going to have to open the device and see how it is wired. You may find that the fix ix as simple as getting rid of the switch, which may be there only by virtue of a wrong theory, but you may also find that you are making the wrong assumptions.
I believe iPhones, and I suspect Garmin satnavs, will not accept a charge from an inadequate current source and you can be sure the solar cell will be inadequate by itself, hence the battery, but I can’t see a problem with getting a charge from the battery while the solar is still connected.
But I think you might find you don’t actually need to do anything, the switch is only there to ensure that all the power from the solar panel is going to the battery, even if the phone could actually do with some, i.e. it is not a matter of must but a matter of preference. This is not such a bad idea if you are short of sunshine.