Technically yes, a “transaction” is on-network - but one can trade wallets (eg: trading paper keys or “Physical bitcoins”). The definition of the word “transaction” is the important bit here.
You can give me the private key to a wallet, but if I'm not connected to the network to see how much money it controls, that's not much good. Even if I trust you temporarily, at some point a network connection is required. Otherwise, it doesn't have anything to do with BTC.
Edit: Additionally, as someone else pointed out, you could have kept your own copy of the key and could empty the wallet before me.
Some physical Bitcoin products solve that too - but I agree completely with you here. That’s why I made it clear that it’s only -technically- possible, but not remotely practical.