If this received no taxpayer funding I wouldn't care, but when public funding is involved there need to be some checks to ensure no abuse, which Rx Kids doesn't have.
All of the checks you listed would just serve to make it harder for this money to reach the mothers that need it.
In the same way seatbelts just serve to make it harder for drivers to get to where they need to go quickly. It would not be difficult to restrict this to people below a certain tax bracket.
It all comes down to how much public money (that could have been used to help people who need it) you're willing to accept the possibility of being defrauded for one person to get support a couple days sooner.
Means testing often leaves out those on the lower end who straddle the line or get excluded by stupid technicalities. You simply can’t account for every scenario. Excluding any mother means you will exclude someone who needs it.
I would honestly love a national program that gives money to every mother automatically no questions asked.
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https://news.ycombinator.com/newsguidelines.html
I've already discussed this in a separate reply.