The rest of the immigrants are still illegally in the country, no? Wasn't deporting them on the campaign platform that got the mandate? What is "the system" if not this?
If your interpretation of the constitution is that as long as you are "protesting" then you can do anything, be anywhere, including whistle along an ongoing police operation then I can tell you that that interpretation is not correct. If it were, any criminal (not need be immigrant) would say that they were "protesting", while robbing a store or doing any other random actually criminal thing.
Edit: I get what you are saying, too, but the practical solution isn't to keep current illegal immigrants in, as in that case anyone attempting to enter legally should just switch to enter illegally as that is more efficient. (I.e, if there is some country-wide entry rate, then the currently illegally entered have succeeded by jumping the line.)
The above is also the platform that was voted into office not long ago. This does look like democracy to me.