I said incestuous leanings not incestuous - based on what OP said.
It home studies are conducted generally over a number of monthsand would involve many meetings with social worker. Mine certainly covered my single status - they wanted to interview ex-partners and if you had none as a 36 year old I would expect a social worker to probe this more, it would have involved not only written references for at least three people including a family member but all referees would also be interviewed in person.
I don't say that a social worker would have necessarily picked up on her obvious flaws (according to OP) but there were at least three opportunities for others to be even lukewarm about their assessment of her ability to parent which would have put the social worker on notice. All references are confidential.
I had to get into details of my sex life, was asked similar questions months apart in a slightly different way and whilst I'd like to consider myself pretty damned smart, I'd have to be obsessively organised in monitoring my own responses to be consistent in answering them.
Of course its not impossible to get through a home study whilst covering up that you're really not up to the job but in my experience the home study does a pretty good job of testing peoples reserves and fortitude and determination. Of course with family and friends telling the SW that SIL would make a fine and dandy mother it does make SW's job a tad harder.
The issue isn't just jealously of OP, according to OP SIL has always been like this although it is possible that her inability to cope has only increased after becoming a mother in which case as I have said earlier I think social services should be informed of OP's concerns.
As I said much earlier its quite possible the PIL's are so involved because they are concerned about SIL's parenting ability which seems to me to be the right and proper response for the child concerned but OP seems more concerned that her DH/DS don't get more, and no-one much seems to care about another child being brought into the same situation.
This of course presumes that the SIL is indeed quite the nasty piece of work OP paints because of course I have no reason not to believe her.