There's a precedent why I think that people will approve of "universal" income: We have "Familienbeihilfe" in Austria. It's a payment of roughly 200€ per child that parents get. Everyone gets it, regardless of income, so noone is jealous. It's the best way of distributing financial aid in my opinion.
Regarding special needs, in my experience it's much easier to get aid for disabilities. I have a kid with special needs, and it's incredible how helpful people are. Social workers are calling us to tell us what kind of aid we should apply for and help us fill out forms etc. It's completely the opposite experience of applying for unemployment or for aid with rent payments etc.