Your simple numbers are, frankly, meaningless and stupid. They assume that childcare is the
only reason poor people are ever out of the labor force, for starters.
If you assume thatpoor people are more likely to have to resort to mutual aid outside the formal economy for that while non-poor people are more likely to be able to be either consumers or suppliers (or both) of childcare in the formal economy and that is the only factor affecting differences in labor force participation between the groups (which is, no doubt, an oversimplification), then you'd expect -- no matter what the mutual aid ratio is -- a lower labor force participation rate for the poor.
If you more generally assume that this is hardly unique, and that regulatory costs force those with less means out of the formal economy in other ways, you'd expect the labor force participation rate of the poor to be even further depressed.
> Can you name any piece of evidence or measurement which would cause you to believe the poor are not hardworking?
I'm not convinced that the character trait of "hard-workingness" has a good objectively-measurable proxy, but then since I'm not the one making a positive claim about it, its not my obligation to present and justify a valid measure of the trait.
> Or is your doubt more religious in nature?
False dichotomy much? Thinking you are making a positive fact claim about something for which no good objective evidence exists and which, while it might in principle be measurable, it is unlikely that objective evidence could exist in practice, isn't a "religious" objection -- if anything, its skepticism that you are making what is essentially a claim of a religious nature (that is, one that is held as true statement of material fact about the world without being either objectively justified nor even necessarily practically objectively justifiable.)