I’m sure you can see that term is intended to portray the person in a specific way.

No, it’s just a normal term, intended to convey that the person is an immigrant, and that they didn’t follow the legal process. Those are just the basic facts about that person’s situation with respect to the law. The legality of their immigration status is often the focus of the discussion. For example, it’s easy to take advantage of illegal immigrants because they might be afraid to go to the police due to their immigration status.

The term has been in use for almost 150 years. Sometimes clinical labels become pejorative over time. But, it doesn’t seem like that’s the case here. A right-winger is much more likely to say “illegals” or “illegal aliens”, if not just using some slur like “wetback”.

Alternative terms that have been proposed are much less precise. For example, “undocumented migrant” is horrible. Not only is “migrant” less specific than “immigrant”, because immigration is a subset of migration, “undocumented” is much less accurate than “illegal”. Most people in a country illegally have documentation, they have passports, birth certificates, sometimes even local driver’s licenses. The issue isn’t that they lack documentation. The issue is that they aren’t following the laws related to migration. Others like “undocumented noncitizen” or “undocumented American” are even worse. What does “American” even mean in that context? Is a fiercely loyal British person who is legally in the US on a work visa a “documented American”?