Neither "open" nor "shut" are actually possible options for schools now.
Even in a hard lockdown, schools are going to stay open, as they did in spring, for vulnerable children and the children of keyworkers. That might only be (depending on the school) a small fraction of children, but it still requires multiple staff, for safeguarding and security. And teachers will keep working supporting remote learning.
Even if schools open fully, it seems unlikely that many children would get a full term of face-to-face teaching. Our school was lucky and had just one reported case, right at the end of the autumn term, but some children still missed weeks of school waiting for (negative) test results. Other local schools had multiple year groups out for weeks, sometimes several times. Some had to reduce provision because of ill or isolating staff. Given the much higher infection rates going into the spring term, it seems likely the most children would get might be three quarters of their normal classroom time, with unplanned switches to and from remote learning.
And there are options short of a full opening. Running with half-classes would halve the chance of an infected student being in the class, allow greater classroom spacing, and roughly halve the potential size of transmission events. The division could be done by alternating weeks, and possibly those children and families who are best placed for remote/home learning — with good tech resources, enough time to juggle other commitments, the ability to provide academic support, etc. — could give up their places to children and families at greater need of direct teaching.
Overall I think it's better for schools to reduce face-to-face provision now, with a plan for changes to allow for progressive opening up, than to reopen without any changes at all and just hope. Whatever happens, schools need to be properly resourced to make their staff and students - and the community - safer, and to help make remote learning work as well as possible for everyone. Attempting to micro-manage this seems well beyond the competence of the Department for Education, and out-sourcing would involve significant losses to corruption and extra layers of management, so I suggest something simple like a one-off per-child block grant to schools. The government needs to stop talking about how important schooling is and put its money where its mouth is.