Free regulatory alignment to the entire UK/ROI has been mooted. The EU are even happy to go along with it. As they are happy for the entire UK to be included in the customs union/trade area/travel area/etc etc.
In fact, that's what the UK wants - they want to stay in the Single Market, the Customs Union, the Free Trade Zone, all that type of thing, but also be free to have Trade Deals with other countries, and do all that while restricting how many EU citizens can enter the UK.
Basically they want to have their cake and eat it, and people like J R-M and even May themselves have said or implied that the EU should let them, and that anything else is an "unreasonable demand" from the EU.
Scarily, many people in the UK seem to believe this, and they too are blaming the EU (and now Ireland) for the impasse.