I can see myself agonizing over something like this, for sure, but if you step back a little, you will see that you have a very, very first-world kind of 'problem' on your hands. Whatever you do, you can't go wrong. Not really. Not at this age.
When our kids were little, private was out of question, so they went to a local primary until ages 8 and 11. Looking back I think they would benefit from better quality of teaching and care from an earlier age, but then again, during their primary school years kids still have a chance to catch up, fill the gaps, and make the move to a more challenging environment. This opportunity becomes very slim in secondary school, so it's crucial to make the right choice at age 11.
Our experience in primary school was good. Teachers were great, nothing ever went badly wrong in terms of studies, but kids were never pushed beyond their comfort zone. It was all very relaxed and we compensated for it with lots of after-school clubs and activities. Kids had friends from a variety of backgrounds, so when I took my DD's friend swimming, for example, this girl commented that her mum has bigger boobs than me
(this is at age 6). That kinda thing... Different priorities 
However, my point is, both state and private give children something of value, and until about age 8 education is more about exploring the world, then some kind of intense training that can only be delivered inside expensive buildings with wonderful facilities. Both options are ok unless you run into some serious problems.