Why are old republicans against immigration?
I don’t think Republicans are more likely than Democrats to support expanding social security.
A simpler, and more likely, explanation for the question at hand is that older Republican voters don't like people who don't look like them and who don't speak their language.
You are just making up an answer that matches your priors. If you want to know why older republicans are less likely than younger ones or older democrats to support legal immigration, you should probably ask them. My guess is you will get all kinds of answers, and if you study the data hard enough we will probably see some degree of tribalism (in all groups, not just this one).
Warning: This is the longest comment I've ever left under someone's Substack, but I hope it helps address your question.
If you understand Republicans to be motivated by concerns about marginal cost to their income, their increasing opposition to most mass immigration after largely going along with it for half a century makes more sense. The party opposing Republicans today is composed of more naturalized immigrants and their American children than at probably at any point within a Baby Boomer's adult lifetime. Democrats today are not Bill Clinton's party so much as Harry Truman's. So in a highly competitive 50-50 system, recent naturalized votes will be like any other marginal votes; under further political hardball to appeal to or deter in the future, with a desire to win elections.
However, the center-left transformation to Truman's party is best understood not through its demography but its stated fiscal agenda. Today's Democrats are promising a highly inflationary agenda in their party platform, which the Manhattan Institute fellow Brian Riedl noted is at $11 trillion in ten year spending versus $2 trillion in revenues. This is unprecedented as their party platforms go for the last two decades, every single one of which proposed revenue items to fully finance spending items before the 2020 election . Accordingly, 2021 witnessed an enormous demand-side stimulus in the form of ARPA, which took existing inflation to come from COVID and made it considerably worse. Negotiations for further front-loading the deficit with more stimulus were only narrowly avoided in the eyes of some Republican voters. Few doubt if Democrats acquire another trifecta that they will attempt more forms of distributing spending to the many interest groups in their coalition without sufficient revenue.
Did immigrants and their American-born citizen kids personally tell the Democratic party to do this? No, of course not. Any mass party aggregates its platform through a complex sets of interests, rather than one set of marching orders. But if inflationary macroeconomic policy is more possible today (just as it was when mass strikes took off after WW2 under Harry S Truman), anti-Democratic party voters will seek ways to discipline a party that is aggregating their platform in this demonstrably harmful way. The marginal shock of reducing immigration into urban labor markets is of little material concern to Republican voters, who skew suburban and exurban. The marginal shock of reducing a future constituency that generally leans Democratic looks more favorable to them (and this lean has been the case for most of American history, arguably starting with Martin Van Buren's campaigns in New York to European immigrants. )
This contrasts with urban centers, where many of the native-born vote Democratic on rational grounds of urban amenities and rising property values not unlike liberal voters in Canada for Trudeau. If those amenities and property values dramatically changed, many would stop voting Democratic. This speaks contrary to another example you gave; wealthy Democrats have rationally assessed that Democrats are not yet raising national upper income brackets like they did in Bill Clinton's day. A multi-millionaire incumbent homeowner in California has seen an enormous growth in personal wealth that easily counteracts California's tax rates or regulations under a center-left party. The same is true of billionaires who experienced the tech industry boom in a small-p progressive state. Why not support more environmentalism? Server farms and FAANG campuses don't face the same air regulation and union costs of car factories, costs which often motivate businessmen to vote Republican.
Finally, Medicare and Social Security are largely projected to be financed in their growing form through enormous budget deficits, not immigration . The number of immigrants required to fulfill revenue for even a quarter of these growing deficits is probably far higher than the number NGO's and Dem party groups have tried to usher in through asylum and parole executive levers to enormous unpopularity under a more anti-immigration Congress. Any actual solution to Medicare is going to come through bipartisan action to cut its promises and require wealthier recipients to pay more. It's not clear Republican voters like those ideas, but it's also not clear Democratic voters do either. Democratic politicians are wisely aggregating their coalition's preferences and understand the future Medicare curve as a piggy bank they can leverage; they have cut it twice now to use for partisan bill's CBO scores . That's good news if you're a Democratic interest group, but bad news if you're a future middle class tax-payer. Democrats do not base their coalition on middle class taxpayers and have avoided raising taxes on the ones who do vote for them for a long time.
It's true some voters are driven by prejudices, either in favor or against immigrants. But like with most policy areas, American voters are somewhat rationally reflecting their self-interest and voting accordingly. The evidence voters in a popular mass democracy are *terribly irrational* is just not very strong, and in general is often exaggerated by people who struggle to make sense of democracy  just as I struggle to make sense of French. I don't hate the French language and I don't think smart libertarians hate democracy. We just don't get it. But that doesn't mean others aren't fluent. Republican politicians are now seeing a situation where there's no loss with Latino voters to oppose further legal immigration, arguably some gain, and are taking it. Their voter's interests were driving them there, but now there's a shift to go further. So it goes.
 page 85, https://media4.manhattan-institute.org/wp-content/uploads/BudgetChartBook-2023.pdf
 Chapter 1 "Creating the Democracy, 1820-1848", https://www.amazon.com/What-Took-Win-History-Democratic/dp/0374200238
 page 55, https://media4.manhattan-institute.org/wp-content/uploads/BudgetChartBook-2023.pdf
“Old conservatives oppose immigration” isn’t specific to the United States. It’s how politics work in pretty much every country. So to explain it, you should probably not rely on US-specific things.
To me it seems almost inherent in the definition of “conservative”. You are conservative when you oppose change, and immigration is nearly always bringing change.
The financial considerations seem secondary.