Isn't he probably right ? I mean, instead of setting up a full node can't we just use the most trusted online website for smaller amounts of bitcoin ?
He WAS right when he said that back in those days because there was a very limited offer on wallets. You had Bitcoin Core that required the download of the entire blockchain. Some people didn't want to do that, because they either didn't care too much about bitcoin at the time, they simply didn't want to, or they didn't have or were prepared to make available the storage capacities to host full nodes. Since light wallets were nonexistent, you only had a chice between exchanges - web wallets - Bitcoin Core.