OP, regarding your environmental point.
Yes it is silly that people send their kids to schools that require car trips when there are schools within walking distance available.
But think, if you lived right next to a school, but had to drive your child to a different one miles away, because next-door-school was full of siblings?
That environmental point is very much in favour of distance as a priority over siblings.
If you don't want to drive to school, you don't move further away. If you do move, you change schools to a new, walkable one.
In your situation, the 'ideal' solution is not possible, because the space you were hoping for for your younger DC rightly went to a child who lives closer to the school, who might otherwise have had to drive to a different school.
So you have to weigh your priorities.
Not moving older child vs not driving/not having them at two different schools) vs not moving house. You can have two of the three. (Probably - if you move house, back close to DC1's school, you might still not get a place, although you would move up the waitinglist).
So which ones matter most? How bad would it be for your older child to change schools - if the benefit is that you can stay in your current house, have both children at the same school and can walk there leisurely together each morning rather than ferrying everyone about? How bad would it be to move house, back close to the older child's school, if the benefit would be that older child does not have to deal with the upheaval of changing schools, and hopefully you get to have both kids at the same school and can walk leisurely in the morning rather than rushing around in the car? And how bad would it be to continue as you are, dealing with two different schools and having to drive each morning, if the benefit is that you can stay in your current house and older child need not change from their current school?