The agrammatical headline gives the very false impression that half of SNAP recipients are non-citizens.
What it actually says is that more than half of non-citizen households (54%) and still more illegal aliens (58%) used food stamps. And even here, the tense is misread: it does not mean that they are currently using SNAP, but that they have used SNAP. This is a rate much higher than native-born Americans (38%), but if it seems odd to you that so many native-born people use foodstamps, only about 13% CURRENTLY use food stamps. This is how many people EVER use food stamps.
Still, the main point — that illegal aliens DO get food stamps because they have CHILDREN who are eligibe — is a very good point to consider. But leading people to believe that SNAP delays affect chiefly non-citizens is simply misinforming people.
Frankly your clarification does not make it any better.