This is only my opinion.
Just like a person needs to give ma’aser with his money, he needs to give with his time etc. That would mean a person should actually do a chessed by himself, not just pay someone who will do the chessed for him. Stories with gedolim lead me to believe a lot of tzedakah being given includes not only the actual money, but also the feeling a person has when giving it. Are you doing the mitzvah of giving tzedakah? Yes. But at the same time, maybe theres a way to do it a little better…?
My suggestion would be to leave the banking system in place but maybe stop the automatic tzedakah payments. Aside from not having to do most of the payments (which automatically is a plus), you get the chance to do this mitzvah in the optimal manner. The famous story with the Vilna Gaon’s wife is coming to mind.
Again, this is just my opinion.