The Layering phase is when the original source of money is concealed through transactions or bookkeeping tricks.
Tornado Cash doesn't move the funds between financial systems at all
"Layering conceals the source of the money through a series of transactions and bookkeeping tricks"
The source of the coins is in plain sight - it's clearly from Tornado Cash, and when converted into fiat currency the source is very likely also obvious (some exchange) unless you do some in-person exchange of cash for coins, in which case this has nothing to do with Tornado Cash
"Integration, the now-laundered money is withdrawn from the legitimate account to be used for whatever purposes the criminals have in mind for it"
Again there's no legitimacy for the coins after going through Tornado Cash and the fiat currency is still under scrutiny.
If you just go dirty wallet -> clean wallet -> exchange it’s trivial to see that the exchange coins came from the dirty wallet. If you mix the coins from the dirty wallet to the clean one there’s no way for the exchange to say the coins came from the dirty wallet, all they know is the coins came from the mixer.