Not sure why you've chosen to go off on a tangent about immigration to the United States in response to my post about populists of all stripes typically trying to remove all sorts of checks on their power, including the democratic ones in the guise of "protecting" the people from the enemy
du jour.
But since you've decided to make this about the current US administration then yes, it's a matter of fact that the current system is a republic with limited democratic participation in the decision making process. A situation in which a single individual holds all the power and permits bureaucrats like ICE to do what they wanted to any individual for any reason with noone else being able to intervene would also be a republic with limited democratic participation in the decision making process, though not one I would prefer, particularly not if I was a law-abiding citizen who understood how legal processes worked but had the sort of ancestry ICE and Trump seem a bit suspicious of...
No systems in the US look like what people campaigned on and voted for. What would policy look like in with true public votes on anything and everything? Well judging by approval ratings, Trump's decision to singlehandedly cause a recession with his tariff policy is something the public overall do not back, no matter how much Trump claims that he's acting for them and against globalist elites. Similarly, it seems that what Trump's newly created bureaucracy is doing to other government departments doesn't meet with wide approval, no matter how much the world's richest man appointed to head after giving him lots of money it claims to be tackling elites and corruption. Trump is a populist, but like many other populists he and many of his actions are not at all popular at the moment: it's a feature of the design of republics rather than the popular will that he remains in power nevertheless. Perhaps that's why the executive's power is supposed to be checked...
But yes, the one area in which his approval ratings stayed above water in some polls is in his handling of immigration. Whether this includes every decision the unelected bureaucracy that is ICE makes to select random non-citizens and citizens for detention or even deportation to to foreign concentration camps for wrongthink or wrong tattoos is another question; seems like public opinion sides with the courts rather than the admin on deporting a legal immigrant for no reason and then not bringing him back because that would mean admitting error. In a republic like the US it's actually not the job of the official elected in the belief he'd get the price of eggs down to overrule courts, and it seems hollow to claim it'd be more democratic to skip the process of legislative change in favour of executive fiat when the public doesn't agree with him on many of the details.
It would, of course, technically be more democratic to have people's right to remain in the country based not upon law but upon the prevailing fashionability of their skin colour and surname with a wider public that got to vote on mass deportations to third countries whenever they felt like it. Even as someone with reason to be completely confident this wouldn't jeopardise my right to remain in the country I was born in, I'd hesitate to say that supplanting citizenship law with a public popularity contest would be better though it would be more democratic. But this has nothing to do with what is going on in the US, which involves the president insisting his his yuge popular mandate places him so far above the law that bureaucrats whose actions he favours don't have to follow it, never mind other populists like Maduro or Mao or Mussolini who didn't even worry too much about the mandate bit.