Paul Krugman used to say that while the Democrats were ambivalent about immigration, the Republicans were schizophrenic. Put in my terms, he meant that the PBPs see immigrants as a needed boost to the workforce, while the Reactionaries loathe them for both economic and cultural reasons. Since the Reactionaries currently call the shots in the GOP on every issue except taxes, their position is the default within the party–hence, the ultimate failure of the immigration legislation during the Obama years.
The economic case for increased immigration under present conditions is clear and overwhelming. Even a reasonable Reactionary would see some benefit in, among other things, providing more child and health care workers and shoring up Social Security. Is that likely to change their position? Do they really want white Christian American farmers, just to name one group, to struggle getting workers the way British farmers do after Brexit?
Alas, the answers to those questions are no and yes, respectively. The GOP will only become reasonable on immigration if its extreme right wing is crushed in an election.