Trump outlawed legal visa holders from Iran from entering the US when he signed the first immigration executive order. I don't think Trump voters were against that.
If you are only opposed to the illegal part and not the immigration part, it's relatively easy to fix: just better align the supply of visas with demand. There's a number of obvious things you could do to achieve that:
(1) Keep by-category caps but eliminate per-country caps within each visa category entirely, or
(2) Keep existing caps, but allow immigrants qualified in existing immigrant visa categories to pay an (initial and annual) additional fee to skip the line and get a "supernumerary visa".
(3) Allow people who aren't barred entry but who aren't qualified in a y existing immigrant category to pay an (initial and annual) additional fee to enter on a special visa.
Obviously, if what one has a problem with is immigration and the "illegal" part is a smokescreen, then one will be less likely to support reforms which address illegality without aiming to reduxe overall immigration.
It's important to judge someone the most by what they do, and not just by what they say.
I don't know a single person, and I know a good number of Trump voters, who is opposed to legal immigration. What many people, myself included, are opposed to is the illegal part.
Yet I consistently see this stance twisted to mean opposition to all immigration. This conflation is patently ridiculous but it makes for a nice straw man.
(edited for clarity)