In fact, in the linked article Neil Leitch makes the entirely sensible point that that using school nurseries to offer childcare for very young children “should only be done where the environment and provision is suitable and of sufficiently high quality and appropriate to their care and development". In other words, the question is not about schools vs private nurseries, but about whether the early provision being offered to two-year olds is appropriate for their needs, or not.
It is true that the discourse of the current discussion about schools admitting two-year olds is largely about finding a cheap solution to the childcare shortage in England. I would argue that "childcare" itself can be very limited in its conception, implying a utilitarian service enabling parents to go to work. Instead, I would argue for early childhood provision that is in the interests of the child, as well as making it possible for parents to balance bringing up their children with going out to work.
The debate about quality for two-year olds is too important to get sucked into the long-running and pointless war of attrition between the different organisations which offer early years provision. When the sector is divided, we are easily defeated: we should be standing together for quality.