It may depend on Izakayas. In my opinion, rather chain Izakayas are children friendly. They have separate smoking and non-smoking seats and seats are Zashiki style (Japanese style where you sit on the floor). Authentic & historical Izakayas where there are no English menu tend to have only smoking seats and menus are thought to be good with alcoholic drinks, so there may not be many options for children to eat. It may be safe to ask the Izakaya you are interested to directly.
It's totally OK to bring 6 year child to Izakaya. Actually, there are many Izakaya restaurant where prepare menu for family and child. The one thing you should concern would be some Izakaya doesn't separate the smoking area and non-smoking area, so your child might get the cigaretttes' smoke.