The problem with super high fees nowadays I'd getting worse and worse. Its understandable that Coinbase and Xapo have to stop paying fees because they're getting horrendously expensive.
Moreover, another problem with using a 'wallet which is free for transactions' is that the confirmation times can vary incredibly. It can increase incredulously. I would not suggest you use such a wallet. If you are using a wallet, like Coinbase, make sure to add an appropriate transaction fee.