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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I mean, where I live there's food banks, and the only requirement to utilize the food bank is that you bring in a piece of mail with your address on it so they can know you're at the correct food bank. I know, I'm poor and I've had to use the foodbank, recently even. Besides that, you could be the richest guy in town and go and pick up a nice big haul of food every 5 weeks if you choose to. But for whatever reason, this system isn't rife with abuse, even though I'm not drug tested (I would pass) or background checked (no criminal record) or made sure I work (I do). I'm aware of a few bad eggs who don't "need" the food bank who access it anyway, but by and large the people who do need it are getting it and that's what matters.
I guess my point is, you always hear about "what if this is abused" when it comes to systems that are in place to help the less-than-fortunate, but from what I can see it's always just a veiled attempt to make it harder for people who need assistance to actually get it, for little to no actual return because the "rampant abuse" we're told to worry about doesn't seem to be such a big issue in reality.