"And while that might be good for the government in terms of appeasing voters fixated on immigration / free movement of people, [...]"
It seems like all the non-EU countries that the Government is currently trying to line up for post-BREXIT trade deals have one common demand. They want their citizens to be allowed freer immigration to work in the UK.
The Tory minister Priti Patel summed it up in the run up to the referendum. She wanted us to leave the EU - so that we could replace EU immigrant workers with more from the Indian subcontinent. Her reasoning was that historically the old British Empire countries had more rights to that privilege than did EU countries.