To an extent, yes — but like all humans, voters can sometimes focus too narrowly on the short term, and/or on their own desires and fears; sometimes a broader perspective is needed. (That's why some elected officials used to say, in effect, "My voters elected me to exercise my judgment.")
A good government should explain to the electorate the consequences of restricting immigration, because in reality people who don't want immigration probably wouldn't actually like living in a country with no immigration. However, it's a lot easier to appear to be doing something, without really doing much.
The ultimate example would be the Brexit vote in the UK. The population were promised that it would solve all their problems, enormous amounts of anger which could only be quelled by an exit were successfully generated and in 2016, 52% of the population voted for to leave. Fast forward to 2023, and only 33% of the population thinks it was a good decision.
There are always a lot of arguments about immigration, that it is useful for skilled people and that they pay taxes. But it does not even pass the sniff test, a local person would pay even more tax (most likely) and needing to import qualified skilled people is just a sign of failure somewhere in your education system. For many reasons you should not want to import skilled labor, it is not a good deal for a country...
However, it is good for-profit focused business to increase competition and lower wages in their industry. Which is exactly what Apple does and others did before. But the same rich people who buy their expensive hardware and have truly little competition in their luxury ivory tower will tell you otherwise. I do not even blame them, they cannot relate at all, they don't have the relevant experience. I feel like some minimum wage hard work with shitty hours should be mandatory to be allowed to talk about those type of things, otherwise it's just people talking out of their asses...
I believe that what are considered elites got there not by being competitive but mostly in ways that are closer to what cheating is. We are in an era of major stagnation in mobility for the lower part of the society precisely because the elite have been reproducing themselves and have shut down the competition by faking it and creating all kind of "standards" impossible to meet for peoples without the means. I think immigration is just another cheat to prevent equalization. If we look at the distribution ok skills/competency of humans, it mostly follow a normal distribution, yet revenues are really not distributed like that...
Shouldn’t the immigration system—and every system—be designed to pacify voters?