Yes, the advice is entirely wrong. ICE have been kidnapping and detaining people who had proof of citizenship on their body at the time.

Anyone likely to be targeted by these thugs needs to be talking to local activist groups that know exactly what's happening first hand.

Offering armchair views that are clearly from a position of ignorance is yes, provable negative value.

Please just don't.

Edit: talked with a friend who's doing some activist work on this in his area. The advice is actually to avoid carrying your passport or similar, as if they detain you they'll just throw it away along with the rest of your belongings, and you'll have to go through the weeks long process of getting a new one.

And that's the happy case if they throw you in the van and let you out hours or a couple days later.