That treaty isn't relevant to the issue of illegal immigration. It's intended to deal with legal visitors from a foreign country. We want the illegals to disappear. Not in some sinister plot to kidnap them but to go home. Again, do really think we contact their government every time we send one back? What's to talk about besides giving them information they could use to make their next invasion attempt successful?
Sorry, the VCCR makes no distinction regarding legal vs. illegal immigration. That's to keep the host government from simply revoking a visa and saying "he's here illegally, so he's no longer under the Vienna Convention."
Again, do really think we contact their government every time we send one back?
We send a summary report saying "So-and-so detained on 5/5/06, deported on 5/8/06." That's the sole requirement for immigration-related offenses.
When we detain and hold them on a felony charge, that's when the requirement to allow consular visits comes into play.
Remember that EP-3E that got smacked by a ChiCom MiG 5 years ago? Technically, the crew didn't legally enter China, and the ChiComs were making noises about charging them with murder. By your standard, the ChiComs could've said "screw you, pal" to requests for consular visits, and we'd've never seen them again--and we'd have exactly zero argument.
If that's what you want for Americans, say so plainly. Just remember that whatever rules you seek to impose on foreigners will be imposed on your fellow Americans.