People are saying that the mother is a disgrace because she is allowing her very reasonable desire to share caring responsibilities to impact on the wellbeing of the child, which is not acceptable.
The OP said that Social Services are involved "because of the arguments" and because of a referral from the child's school. Schools do not make referrals to Social Services, and Social Services do not get involved, when a child's parents are simply exchanging emails about contact arrangements.
The OP has also stated very clearly that neither parent wants to live with the child full time and, as a concerned relative, she is now asking questions about foster care.
Perhaps you don't find any of the above at all concerning and the Social Services involvement/relatives discussing foster care options is all part of your "normal". However, for many people, those aspects of what the OP has shared are massive red flags that indicate that neither parent is prioritising the wellbeing of the child here.
Personally, I think it's incredibly naive to assume that a 10yo child won't have picked up on any of what's going on in this situation. They will almost certainly be aware of at least some of it, and that's heartbreaking. But even if we set that aside and assume that the child is blissfully unaware of any of what has gone on, what kind of parent would want their child to spend half of every week living with someone who has made it abundantly clear that they really don't want them? I couldn't do that to my child, could you?
Asking the child's father to step up and do his fair share is fine. Allowing the child to suffer the consequences when the father fails to do so is not. And that is why people are saying that the mother is a disgrace.
The father is a disgrace too, of course. That goes without saying. But two wrongs don't make a right, and ultimately, in this situation, both parents are failing an innocent child. You can defend that all you like. I think it's abhorrent.