I do agree with the parent -- to trade BitCoin, it can be purely physical (as in from bitcoinpaperwallet.com). However, if Alice gave a piece of paper with a wallet's private key to Bob and the identical one to Charlie, the first one who gets the transaction in the system and through the blockchain gets the BTC, the second is screwed.
True, you should trade physical bitcoins with people you trust.
As for the blockchain, there are supposedly shortcuts (mainly letting people run it for you...) but same problem with those as with exchanges. Said group can easily change results in order to snag your stuff or delay it enough for a double-spending attack to take place.
This applies to most online wallets(not all) , somewhat with SPV nodes but not really , but nothing to do with pruned nodes that are around 1GB in size.