An equivalent with your driving example would be "illegal speeding" (as opposed to, say, an ambulance, which can legally be given permission to speed).
The big difference is that "illegal alien" really does refer to a person simply existing in a country they are not supposed to be in, rather than an action like speeding, hence why shortening it to just the adjective "illegals" works. It describes a state of being, just like "asian" or "programmer".
Correct, I have no issue with the term: "illegal alien"
> shortening it to just the adjective "illegals" works
in a derogatory manner
Americans don't, typically, describe people who received a parking ticket, or a speeding ticket, as criminals. In the same way, most people have not, historically, referred to people overstaying or working while on non-work visas, as illegals.
Melania Trump makes a particularly good example for this. Her profession as a model is high status, so the allegations that she worked without an appropriate visa in the US in the mid 90s does not generally lead to people calling her a criminal.
Case in point, I have a young lady on my team at work that came to the US to receive her masters in computer science, and is presently employed on a STEM OPT extension to her F1 visa. We have tried twice to get her H1-B and she hasn't been picked up. Next year will be the last try, and we're unsuccessful she will be kicked out of the US to no press or protest. She's a wonderful person and has been a true asset to my team and the company, she did everything 'right' and yet may have to leave. However if we do the exact same thing to someone that skipped all of those formalities and broke the rules, it's somehow racist and nationalist and practically a human rights violation?
I get it, calling these folks 'illegals' is a bit crass. But it's more precise than calling them 'immigrants' because it's a special case of immigrant that we're talking about, one is distinct not because of race or color or country of origin, but because of legal status.
If you are still here to be deported after your time is up, you haven't followed all the rules, since leaving no later than when your time is up is, itself, a requirement of the rules.
On the other hand, while people who are concerned with the recent border crossers being denied the right to apply for asylum, or being held in humane conditions, may not be concerned about those visa overstayers who are not facing either of those problems, the people who are concerned with the leniency for those illegally present who have done nothing seriously wrong beyond immigration violations tend to be just as well inclined to visa overstayers as any other person illegally present; the distinction you suggest is not accurate.