That's not quite the "gotcha" they imagine, Reagan was to the "left" of his own party* on immigration, they were condemning him for the "amnesty" quite fiercely already by 1988. Pat Buchanan's entire insurgency was motivated by it. H.W. Bush's "betrayal" with NAFTA and on the taxes pledge was fuel on the fire. (Also Buchanan didn't believe he could legitimately run in 1988.)
*And the Democrats, the Unions still had a say in the 1980s Democratic Party.