I am very skeptical about "paperless" banking. I understand why banks want it, but the advantages all seem to be for the bank not the customer.
You should always keep records of your finances. You might need those records in the case of a dispute. You might also need them for other reasons, such as applying for visa's to travel (though granted this is more of an issue for people who live in poor countries), filling in a tax return, or convincing someone that you are not engaging in money laundering when you move money around to make a big purchase such as a house.
Trusting the counterparty to keep those records seems like folly to me. It's unlikely that a bank will deliberately falsify the records, but it is far more likely that they could become difficult to access due to some reorganisation. Or you will have a problem logging in and they will ask you for information about past transactions as part of their process to verify your identity.
With paper statements the company sends you a statement, it sits in your pile of mail
until you get around to filing it. If the company gets bought out or re-designs their website you still have that paper.
If financial institutions would send statements by e-mail I wouldn't mind so much, since I control the achiving of that e-mail, but most financial institutions nowadays consider e-mail to be insecure. So they put the messages in a secure inbox that is locked behind the online banking login.
Yes, you could diligently log into online banking and save or print copies of the statements each month, but life happens. Will you really be diligent about doing that every month?