I have been using Coinbase for over two years and have never had any problems with them. I make transactions via Coinbase about once a week.
The only issues I've heard of are when someone uses their addresses for casino games or sports betting.
In my opinion, this is the safest exchange on the market, but of course I never recommend keeping large amounts on web wallets.
I've used Coinbase too. If you keep anything there that you can't afford to lose, it is risky. It may be called the safest exchange, it has like 98% of their funds in cold wallets, but keeping funds in any exchange is risky. I wrote this about Coinbase specifically because it is the most widely used wallet for newbies. Anybody can easily set up an account, deposit and start using it instantly without any backups or so. It doesn't offer the keys of the addresses too. So basically, all your funds are in their hands. Why would anybody blindly trust them? I didn't have issues until the number 3 happened.
Of course, keeping funds in the wallets to which you do not have a private key is extremely risky. The same as keeping funds in wallet to which someone else has a private key (not just you). In both situations what happen with funds depends on the honesty of other person.
Coinbase was created for less experienced users, which is why it is probably the most user friendly on market.
When it comes to delaying transfers, they do so for security reasons. They are simply afraid of being accused of money laundering. It is possible that they do it too carefully.
And what about the app part? Why does the transaction go through on the app and not on the website? You can see there's something going on here.