You're so right about how horrible it is that this has fallen into the culture wars. That's one of my two big frustrations. The other one is that THIS AIN'T BRAIN SURGERY!!!!! Other countries have figured out how to do this: they have good ideas that we could just copy, if we would just climb down off our barricades and talk. " If I were Grand Poobah, I'd create a bi-partisan panel of immigration nerds and say "go figure it out"." You've absolute got my vote for that.
I live in Brazil. Left and Right reached consensus on immigration here long ago, and while Lula and Bolsonaro disagree on just about everything else, there's no light between them on immigration. It's a non-issue. This country has a more liberal policy than the US on asylum and admitting people on humanitarian grounds, with special concern for natural disasters and stateless people. There are a lot of Afghan and Syrian refugees, a lot of Haitian refugees, and, especially where I live, a LOT of Venezuelan refugees. They're processed quickly and respectfully, for the most part, and have labor cards for legal work within a couple of weeks, if they're determined to be eligible to stay. Brazil follows general international "first safe country" principles: people admitted on humanitarian grounds are free to move about the country and to engage in international travel if they clear it with the Federal Police, but if they leave without advising the Federal Police, they're assumed to be looking for a better deal somewhere else, and to have relinquished their humanitarian status. They can't come back.
People with a Brazilian parent, spouse, or child can request permanent residency. I married a Brazilian in July 2017, in the US. We registered our marriage at the Brazilian Consulate that month, I had my resident visa in August, and I moved to Brazil in November. While Brazil is very liberal with immediate family, it has built-in safeguards in the system to prevent chain migration: people entering the country this way can bring parents, grandparents, and minor children. They can only bring adult siblings who are disabled and totally dependent on the immigrant.
One of the worst aspects of the US system that doesn't get enough attention is the way it abuses LEGAL immigrants. A Brazilian marrying an American here and doing what I did in reverse would probably have to wait at least a year for the similar visa, and be treated with a great deal more suspicion.
Non-family based immigration is based strictly on skills. A potential immigrant needs to have a job offer in hand from a Brazilian employer. This skews non-family immigration strongly toward STEM and other technical fields, if in a rather heavyhanded way. Illegal immigration is controlled through stringently enforced employer penalties for hiring illegals, and does not seem to exist in any meaningful way.
A permanent resident foreigner in Brazil has all of the non-political rights of a citizen and is under no pressure to become a citizen, although s/he may, within one to four years. Unless from a Portuguese-speaking country, the candidate has to prove the ability to speak, read, and write Portuguese at about high school graduate level to be naturalized; the language requirement is lifted after a person has been a permanent resident for fifteen years.
Canada and Australia have a point-system based evaluation system for non-humanitarian immigration that I like more than Brazil's. It gives points for age, language ability in the national languages (Canada), level of education, field(s) of study, professional achievements, and other criteria.
My ideal immigration regime would include an asylum and humanitarian regime more like Brazil's, processing speed like Brazil's, family admission policies like Brazil's, treatment of long-term permanent residents like Brazil's. and a point system for all other immigration, like Canada's.
In the US, there WERE health checks before 1924, which were often applied in a discriminatory way, especially against what was explained as "feeblemindedness". My reference earlier to 1870 was to that pre-1924 system. Whether in 1870 (pop. 38.9 M) or 1920 (pop. 106 M), we were a rapidly industrializing country with a chronic labor shortage that needed vast armies of unskilled labor to staff new and growing factories. We're now a post-industrial country of 331.4 M, and I don't think that massive unskilled immigration is relevant to our current reality. If anything, I think it would tend to perpetuate abuses and inefficiencies that we should be fixing instead.
Over to you.