When my children were young my husband often worked irregular hours and my kids had activities, so we often ended up having meals separate (usually me and the kids - with me joining my husband for tea when he got home for his dinner). I made a point of Sunday dinners - good dishes, extra tableware, serving dishes passed around in an orderly fashion - saying grace, special meal. They learned their manners. They were included in the conversations. They took part. They often recall how much it meant to them.
In much the same way, I tried to teach them about worship in church. Yes, they attend Sunday School. But if there was no Sunday School, they attended worship with me. Gradually they learned the hymns, the Lord's prayer. Sometimes they were asked to help usher, pass out hymnbooks, light the Christ candle, or take up the offering. Sometimes they read scripture. They also attended Ash Wednesday, Maundy Thursday and Good Friday services with me. (So did my grandchildren). And I think they learned what it meant to be part of the church family.
But, there was a time when the church introduced something called Intergenerational Worship. The idea might have had merit, but I never saw it done well. Instead it became a mishmash of various parts that didn't seem to relate to anyone. There seemed to be lots of reading - responsive or group - but too fast, long or difficult for young readers. Often a poorly done play rather than a sermon. Additional music where children would show off their talents by playing an instrument or singing. Often a longer than usual service to fit everything in. My kids didn't like it. I didn't like it. Adults without children started staying away (or sometimes got up and walked out) on Intergenerational Sundays. I don't know what it was meant to accomplish, but it certainly didn't prepare kids to worship in a regular service.
To me this was worse than sitting them at a separate 'kids' table'.
First of all, I am thinking about the supper table - the one where families gather. And a smile comes to my face. At our house we had very limited space. Our kitchen was very small and we had no dining room. At the time my children were little we had quite a large family that gathered on special occasions. The adults would eat around the living room (usually on tv tables) as it was the biggest room. And the children would eat lined up in the hallway (they still talk about this to this day LOL!). It was the best we could do. One of the eldest of the children commented about "graduating" from the hallway to the living room when she became a teenager. They were happy times and we didn't worry about too much about where everyone sat - they were fed, and included in the gathering, those were the main things.
Second of all, regarding church. I don't think we can always include everyone in everything all the time. In saying that, we do try to include everyone at a variety of times throughout the year. On each Sunday our children are present in our worship service for the first 20 minutes and then they leave for Sunday School. If it is communion Sunday, they return later for that part of the service. In a variety of ways we involve the children in learning about church and serving in a variety of ways in the worship service (ushering, serving, reading, etc.).
We have the kids in church for the first 20 min. or so (up to the Moment of Wonder as we call our story time), then they go to class. About 3-4 times a year, RE takes a service which generally has the kids doing various parts of the service (often very well, there are a couple that I would happily have as service leader when I am preaching). As part of our planning for going lay-led, the proposal has been made to have RE take one service a month starting in the fall. And I think that came from a member of the RE committee, so it isn't just someone on worship pipe-dreaming.
In terms of the dinner table - when I was a kid & the whole extended family would gather - we kids LOVED eating at the card table set up in the playroom! It was very special, and a lot of fun. We did not feel excluded, as this was only for the meal & for everything else the whole family mixed together - rather we felt honoured to have our own space.
In my church - things happen much as others have described above. Kids start with the whole congregation, take on various roles (reading, carrying the Christ candle, singing in the kids choir) then leave for Children's Church; rejoin us when it is a communion service.
I am not a fan of 'children's time' at the chancel with the minister - seems to contrived & the kids being paraded as entertainment for some of the old folks - so not very respectful IMO of the kids. Many would disagree, and I get that.
On White Gift Sunday, the kids do the entire service - and it's usually wonderful!