ThatBritGuy:

But for consumers, there's very little practical difference between wage inflation and asset inflation.

Not quite. Assets inflation benefits asset owners who see their net worth grow (along with the income they can earn), while everything a consumer spends money for is not necessarily an asset: food, clothing, transportation and utilities...
Wage inflation rather than asset inflation is actually much more useful for these everyday expenses, and this is precisely where the gap has grown over the past decades, partly covered by growing debt: Assets have grown. Wages have not.