Maintaining the integrity of a sovereign nation is hardly 'trying to have our cake and eat it' - the UK, including Northern Ireland, have a right to leave the EU, lock, stock and barrel. The key problem with Brexit is the fact that the EU require a hard border where there cannot be one for political and social reasons (that had/have nothing to do with Brexit).
As I've maintained all along, keeping the Irish border open is not and should not be a stick for either side to beat the other with (the EU or the UK, I mean)... it is not negotiable that the Irish border stays open, and thus both sides should have been operating under that assumption from Day One. The UK and Ireland have maintained this position throughout, while the EU are merely paying lip-service to the idea because they know that a No Deal Brexit will mean they will have to demand that Ireland impose new border checks in order to maintain the integrity of the Single Market.
What is becoming apparent now is that there are indeed solutions to avoid a hard border, but the EU will only entertain them if they are given no other choice. Short of suspending Ireland from the Single Market, the only other way for the EU to avoid having to impose border controls in Ireland is to come up with a deal that the UK can accept. Thus far they have failed to do so, and hence No Deal is now staring them in the face.
Luckily for them, however, the original backstop idea is now gaining more and more traction - far from 'trying to have our cake and eat it', however, such a compromise would be a massive concession (virtually unprecedented in international law) of the UK allowing the EU to maintain its control/rules over Northern Ireland until another solution can be found.
I've never been massively in favour of this because of the precedent it sets (i.e. allowing different parts of a single sovereign state to have different arrangements with the EU), but at the moment it seems like the lesser of several evils that would, at the very least, resolve the Irish situation and allow the UK to move on.