Well you cite Switzerland - they have done exactly what you are claiming cannot be done. They are not in the EU, not in the Single Market and not in the Customs Union. Their relationship with the EU is based on a very specific set of bilateral agreements. So we see that the EU are open to a fudge or 'cherry picking' as you call it - Otherwise the EU would have done exactly what you suggest and say it is full EU membership or nothing.
Actually the Swiss model was proposed by the EU and... rejected by the UK! Remember David Davies asking for a “Canada plus plus plus deal”?
True, the deal between the EU and Switzerland is very specific but the agreement, at its core, includes the 4 freedoms of movement, ECJ jurisdiction and pay into the EU budget (bilateral agreement I and II). Since 2008, they have also joined the Schengen Area. So, actually this is not a fudge. Is this what the UK want? I shall leave it to you to answer...
PS these bilateral agreements have been developed and matured since early 70ies (EFTA membership); the EEA treaty between the Swiss and the EU started in ‘94. Don’t get me wrong, but i have the feeling time is something the UK don’t have. Looming problems, however.....