I think their point is that CPI calculations use a single basket of goods for 'everyone' while in reality different people

So their example was that if restaurant food hasn't increased in price by much, but basic groceries have then the 'inflation' seen by different groups may be very different.

Another example may be that if average housing costs increase just a bit, but the lowest rents have increased the most then cost of living changes could be significantly different between mid and low income families.

I'm not saying either of those examples accurately reflect how things are changing in the US (as I don't have the data to hand), but they are realistic scenarios.

For most goods, it's the opposite -- over the last decades, luxury goods have inflated faster than staples. Similarly, real incomes have increased faster for higher incomes. At least, that's what the last paper I read on the topic said, which was published about 8 years ago.

I think I've shown some examples that are really tough to represent in the statistics. Please read them again and reply how you think they are considered. E.g. how is it considered that cost of owning a house has increased together with the sell value, while the rent has increased without increasing the quality/value/size of the apartments?

My whole point is that adding things and then dividing by the sum of things is not a way to represent the reality that people face. You need to argue against that if you want to make a counter point. Just repeating what the person said that I responded to (after reading what he wrote) doesn't provide anything to the discussion.