Not your keys - not your coins, that's 100% true but for some people, I would say: Even your keys - not your coins.
Okay, woud love to hear opinion about this from the community members.
As we know and from every logic, we shouldn't trust web wallets and also exchanges. There were a lot of accidents when exchanges were hacked and people lost their coins. Some may say - don't keep your coins at exchanges for a long time but I would say that everything happens in seconds, maybe you had access on your coins at exchanges while you were replying to this thread but once you submitted the post, exchange got hacked and coins were lost, everything happens in seconds, otherwise loss wouldn't occur.
So, how do you trade in these cases? Only p2p exchanges? But it's not flexible and you may miss a good time frames where you could profit 1-2% of your balance.
I've also had the same question as you. I've been hearing not to hold coins (Or even Fiat balance) in exchanges, for safety concerns, I don't see how you can actually do that. Not only that, but I find it a waste of time and money to have both your balance and your coins transferred all the time.
I've always used online wallets in the past (BitGo and Blockchain) and never had any issues. Currently, I have some money on an exchange and some Bitcoin, I keep investing/trading both, back and forth throughout the week, I don't see how I could actually keep them "offline". I think, that if you are using reputable sites, don't sign up on any stupid phising or scam sites, use 2FA, you'll be good to go. I might be wrong though, that's just my 2 cents.