Minor conveniences like this are not worth the complete erosion of privacy, in my opinion. Just go to the nearest ATM to deposit checks (who uses checks anymore, btw?) and use the site for everything else. Not everyone even has a smartphone, and out of those that do, many prefer banking on their laptop over their phone anyway, which incentivizes banks to create feature-rich websites. If the mobile site isn't any good, usually the "desktop site" isn't too difficult to navigate on mobile, if you need to.
>Go for a walk every day, and occasionally make your walk to an ATM?
There are workarounds, but it sounds annoying and a burden. What if the closest bank branch is an hour on foot away? Or the OP lives in a rural place and it's half an hour drive? I don't have this problem since my bank works with graphene, but I would reconsider using it if most applications I use refused to load.