Except ... what determines if this is better or worse is the alternative. If you're really after the welfare of the child, what determines your position is the alternative ... What damages the child more? Taking the child away, in India, leads to this:
https://borgenproject.org/orphans-in-india/
So no, a parent who's a sex worker is a LOT better. And that should be the basis for your moral position. It's all well and good to define an ideal and then blame people for not attaining perfection ... but of course we'd all be in jail if that's how we actually work.
You want to change sex work? Offer these people some better options. And if you can't do that, for whatever reason, throwing the child into child services IS NOT a good second option.
And I would argue this applies to pretty much every country in the world, including the US. The situation of orphans may not be quite as bad as in India, it's certainly not better than living with (very) suboptimal parents.
And, frankly, that should be the only concern.
"But the other job earning less money would be worse for kid!" is extremely one-dimensional view on that
And I'm not sure how your weird rant about child services is even related to that
That's the only alternative you/society offers for these children. So that's what you should compare to. I get that you don't want to do that, because it totally demolishes your argument, but it still is the only alternative.
"My mommy sucks dicks for money"
That's what
It's really not difficult to find a true statement that is also not the whole truth. It's not like my kid understands what I do for a living beyond the company name I work for. Abstraction is fine and normal.
Am I lying to my kid when I say I "work with computers"? No, it's an abstraction. My 8 year old doesn't need to understand what a datacenter is, but just because I elide details doesn't mean I'm lying to them.
But yeah there is plenty of worse things that can happen.