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.
It is an often reminder to all of newbie and even onto intermediate/expert traders that you must only trade and/or engage in trading if the money or capital you would spend in it would not be all of your money or the money that you cannot afford to lose. Every time that someone asks me where to trade, how to trade, etc., I often tell them that they should only trade if they could take the risk of losing their money. With that being said, there's no real fine line that one must trust any wallets nor any exchanges at all. Even I myself still holds money on some exchanges, not to the fact that I trust those exchanges and their system, but for a reason that I already set myself and my mindset that whatever happens, it would be fine to lose those money.
Hence, whatever your trust issues with wallets and exchanges are, always keep in mind that engaging/trading with crypto remains to be a huge risk to take. Yet, it would always depend on how much risk you can take and if you are willing to take a leap of faith.