Apple Pay doesn't know anything about balances that you may have on Oyster Cards. When you swipe an Oyster Card at a barrier it'll show you the remaining balance (IIRC, it's been a while since I cancelled mine). When you swipe a phone with Apple Pay you don't get any balance information, since there is no concept of a balance, just individual transactions debited from your linked account.
You're right that you need to add the same debit card on the TfL website, and yes you need to sign in to the website to view the information. I'm not aware of an official TfL mobile app for viewing your journey history and spend, but there may be third party apps that enable this.