It is normal to worry about sending your crypto and personal details to online exchanges as you lose full control once it leaves your own wallet. While keeping everything offline is safest it stops you from trading or investing. So best way is to manage risk smartly research exchanges well and only keep only those funds which need for active trading and keep rest in secure storage. It is true that using third parties always has some risk so take risk only for that which can be afford. Healthy dose of caution is good but do not let anxiety prevent you from exploring its potential.
And when it's not your keys, not your coin, it becomes very hard for people to accept, then they should know that the control over your coin is no longer there, so for those that have an interest in holding a large amount of coin, you will need to make sure that the priority of your coin, is what you will look at and if you want to be a trader then it will better to keep the one you want to use in an exchange, but if anyone one is deciding to hold then the best thing, will be for them to be in the wallet until it is needed. but for trading, it will be better for you to know the purpose of both the wallet and exchange having that understanding matters a lot.