yes, just like Emirates, Singapore, Quantas, American... etc all have trouble flying into the EU. i dont know why the Remainers keep wanting to play hypothetical "if the EU want to be nasty and force a hard Brexit" scenarios.
neither do i understand or why there is a myth that May, Davis, et al are targeting a hard Brexit. they haven't been so specific, they are saying that if the EU don't want to play nice, dont want to accept any compromises, we'll be prepared for that scenario. its up to the EU how far they are willing to go to be seen to punish our withdrawal.
Yes - your final point is the nub of it. What WE want is not the issue - it's really down to the EU (or rather, 27 national governments, plus several regional governments who also need to agree). I would take issue, however, with your point about us "being prepared for that scenario" (where the EU won't accept compromises). I think we are massively under-prepared for that....as soon as anyone raises the practicalities of a hard Brexit, they are decried as a scare-mongerer. The mantra is that "Brexit means Brexit", and "It's the will of the people". Ultimately it will be interesting to see just how willing the people are to accept all the myriad downsides of life on our own.