Yes, of course it is possible to have that many kids. Outstanding schools are generally heavily oversubscribed. The vast majority of schools use distance as a tie breaker so, unless your child is in a higher admission category (looked after/formerly looked after child, sibling, etc.), you will always lose out to people who live nearer the school.
You chose to name six schools where you had little to no chance of getting a place. You didn't get any of those, so you were allocated a place at the nearest school with places available. Nothing to do with the council "having another plan". That is the way the system works everywhere in England.
If the appeals for these schools are infant class size, your chances of success are minimal. Given the distance to these schools, it is highly unlikely that a mistake cost you a place at any of them. It is also highly unlikely that you will get a place at any of them through the waiting list.
If you really want your child to go to one of these schools, you need to move much closer to one of them. That would move you up the waiting list for that school, giving you a chance of getting a place that way. But, even if you did that, there are no guarantees a place would come up before September.