MPP doesn't stop your peers from getting information on how big your stash is. They know what UTXO's you used and can follow that back.
Private nodes and private channels don't solve that problem either. The likes of Chainalysis are certainly running nodes themselves, and finding ways to buy and steal data from nodes not run by them.
Coinjoin+LN is basic defense in depth. I always fund my channels from coinjoined funds.
As I said: I will send you an amount of sats you want from my private LN node and you will not be able to find any UTXO from where are originated, not my stash, not my identity, nor where I live or from where I acquired those sats.
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