Obviously it's fair to say that inflation may be measured more in some prices than others, but you're sort of muddying the waters by referring to housing as both an asset class and a cost-of-living, right?
Perversely, if inflation primarily affects "assets" (i.e. of the investment/speculation type), the premise of the original question here is sort of flawed: if increasing the money supply increases the price of assets, those with savings get richer—exactly the opposite of the ordinary understanding of "inflation!"
None of this makes sense. What is happening with all the new money that is entering the economy? Is there really someone with a Scrooge McDuck vault out there? Of course, if inflation picks up that vault has no reason to exist anymore and the stimulus checks raised inflation as intended.
(Maybe this is what you're saying as well?)
If the Fed prints $1,200 for every adult making less than whatever it was, and prices of everyday goods and services don't go up, and instead those adults all turn around and spend their $1,200 on Gamestop and NFTs...
...it all kinda makes sense.
https://nymag.com/intelligencer/2021/04/nft-future-of-money....
Perhaps more explicitly: purely quantitative attempts to understand the economy (as with monetarist explanations) fail to account for the social component. Maybe people are bored 'cause of the pandemic, and as a result, they spent more of that $1,200 on gambling as a form of entertainment, instead of on restaurants or new clothing.
Animal spirits.
Which is not inflation, but asset appreciation (i.e. the goal of investment). Inflation measures the price level of consumables. Stock and real estate is not consumed.
I'd be happy to be disproven with data, but my anecdotal experience since 2008 is that while my purchasing power hasn't budged much (for things like electronics, travel, food...), my ability to purchase a house from my wages has decreased and will only continue to do so (despite my wage nominally increasing during that time period).