VAT is the most unjust tax there is. People with low income spend it on things like food, clothing, housing, entertainment, that are all subjected to VAT, whereas richer people have access to the stock market, investment, and tax lawyers, and pay relatively less VAT. VAT is effectively a degressive tax, if you leave only VAT then you accept a system where the poor pay for the rich.
However this is already the case in Europe because the VAT rate is so high. For example for France, tax revenue in 2014 was 74B€ of income tax, 39B€ of corporate tax, and 139B€ of VAT. I'm guessing this is a political game where you have to keep income tax high enough so that the inequalities are not too obvious.
It depends on the country I guess, in France you do pay VAT on food (5.5%), clothing (20%), and for rent it depends on what you're renting but for most people you're right that they're probably exempt.
However this is already the case in Europe because the VAT rate is so high. For example for France, tax revenue in 2014 was 74B€ of income tax, 39B€ of corporate tax, and 139B€ of VAT. I'm guessing this is a political game where you have to keep income tax high enough so that the inequalities are not too obvious.