The inflation basket only represent a hypothetical average person, who doesn't even exist.

It's more useful to construct multiple separate inflation measures that represent different types of people. Like a "typical renter" inflation figure vs a "typical homeowner" inflation figure. It wouldn't be hard to do and would shine a light on inequality and help explain the rise of populism in certain segments of society.

An even better measure would somehow appropriately normalize the figure by the average disposable income in each of the segments to come up with a figure that measures the felt impact better.

The figure would be negative for wealthy people (who actually benefit from inflation because of asset price inflation) and positive for poor people (whose disposable income mostly goes to rent).