But I think the majority hold that if you're here illegally, and it is properly adjudicated, then you would be eligible for removal.

Sure - at that level of abstraction. But if you start giving them the details (mother of three young children who are U.S. citizens, lived here for twenty years without doing any harm, pillar of the community, etc.), they would be reluctant to see the full weight of deportation fall on that particular person.

Should people who steal be "eligible" for going to jail? Sure. Should 100% of every high schooler who gets caught shoplifting be arrested, arraigned, and prosecuted? Probably not. Who should be more vigorously prosecuted - prostitutes or johns? Etc. Our criminal justice system is built around some level of discretion and judgment not just at the law-writing level, but also in the enforcement decisions and prosecutorial choices. It's a recognition that written words in the statute books are unlikely to ever be drafted finely enough to achieve actual justice in the way that a system made up of humans exercising judgment can.

That's why I think a majority of the people who are most vigorously upset with what ICE is doing, especially their choice to go after so many folks who have not engaged in criminal behavior other than their immigration status, are not likely to support a blanket rule that every person who has entered illegally should be removed. Eligible for removal? Perhaps, just like very shoplifter and every prostitute/john is eligible to be tried to the full extent of the law. But I don't think that a majority would think that expelling that mother of three and leaving her young children behind is the right outcome.