I agree wholeheartedly.
For those of us (most of us!) with a background interest in such things, class is a combination of:
Fungible wealth - money and financial assets
Non-fungible wealth - properties, artworks, businesses &co
Education
Employment status
Family background - titles & ranks
Intellectual capital - comes mainly from childhood environment
Social capital - who you know, security of relationships
Financial capital - what you own; includes depreciating assets like cars.
There are some other markers (I used to use these analyses in my work) such as where you live and what kind of holidays you have. In the UK at present, social mobility is very fluid. A growing number of socially-mobile people are in the precariat, regardless of class markers. Precarity is not a good thing but at least it mixes the traditional classes up a bit.