The latest government guidance (updated 16 June) says:
From the week commencing 1 June 2020, we are asking primary schools to welcome back children in nursery, Reception, year 1 and year 6, alongside priority groups. Primary schools that have capacity after making provision available for these groups can invite additional children to return, so long as they can accommodate more children while still following the approaches set out in the protective measures guidance and their own risk assessment
I don't think many people are aware of, or acting on, the second sentence.
The problem I think is that it is right that decisions like this have to be made locally, because schools are genuinely very different, in terms of what is possible. But I suspect that what different schools are offering is much more variable than can be justified on the basis of buildings, staff availability and other factors.
Unfortunately, some schools are offering very little remote education and actively deterring their pupils from returning. Teachers who are working hard are understandably upset that their profession is being criticised for being lazy. Parents whose children are receiving nothing from their school are understandably upset that their children are not receiving the same support that is provided by other schools.
I'm not sure there is any point in lobbying the government, except as a general protest. What would be far more effective I think is parents lobbying their children's schools. I no longer have children in school but if I did and was unhappy with the provision, I'd be writing to governors and pointing to concrete examples of where other similar schools were doing more.