Look, I'm a liberal but I will call out the Dems when I need to. Most of the reforms to legal immigration have always been a sort of addendum, or a two-part deal with relief for the illegal immigrants included in them. Democrats haven't proposed a bill which purely targets legal immigration. So its a little unfair to say that conservatives are opposed to all kinds of Immigration.
Perhaps, but conservatives have frequently put forward bills (or, when in the executive, instituted executive policy without Congressional action) that further restrict currently-legal immigration and non-immigrant entry and privileges, which is more significant sign of opposition to legal immigration than merely voting against expansions to legal immigration, whether pure or not.
Maybe it is just posturing and lies, but the conservative viewpoint has been to have a merit-based system for permitting immigrants into the country, as opposed to immigration based on familial relations. Every country gets to decide what kind of people that they want coming into their country and I think its perfectly fine for conservatives to feel that way (it is a different matter that I don't agree with that idea). My perception (again, which might be wrong) is that that is essentially what Republicans have been getting at for the past couple of decades (perhaps more).
Now, that being said: these are the same folks who were about Family Values, Fiscal Purity etc. So, like I said before, I don't think the arguments are being made in good faith. But the argument does have merit.
Go look up the stuff that Orin Hatch and Rubio sponsored with regards to H1B visas. The stuff that they sponsored would double the number of H1Bs and was majority sponsored by republicans.
further restrict currently-legal immigration
This has nothing to do with debates over executive authority to choose how to prioritize enforcement and to make transparent commitments regarding that (which the executive clearly has in some cases, outside the obvious pardon power, since literally every grant of immunity from prosecution, or plea bargain to reduced charges, is an enforceable commitment by the executive not to enforce some law in some well-defined circumstance based on policy priorities.)
I don't know what groupthink you are talking about, but there is a wife perception that the US has recently become hostile toward immigrants. Which is reflected not only in fuzzy attitudes, but concrete policy, and it does explain changes in immigration, both legal and otherwise.
> There's also the cliche (spread by the internet and political groups) that conservative/Republicans/etc... are against all immigration. This is 90% incorrect
It would be more accurate to say that there is a wide (and accurate) perception that conservatives are generally more negatively inclined to immigration, and that this is generally correct on balance (though obviously not a reliable guide to individual attitudes.)
> Apparently the problem the conservatives have isn't immigration as an umbrella policy. It's legal versus illegal methods
That's what they usually say, but they also usually oppose policy which would make legal immigration easier and relieve pressure that creates illegal immigration, and frequently support policies to restrict legal immigration. And it's not just 10% of conservatives supporting this kind of policy.
This reveals that the rhetoric that it it is only illegal immigration that is the concern is not reflective of the actual concerns.