I'm honestly baffled every time I hear about people in the US spending their time literally paying bills. Direct debiting works and has worked for decades in Germany, and also across Europe since SEPA. Same goes for savings and transfer scheduling, no problem at all. Doing "if below X, do this otherwise don't" is not something easily done with bank's own software, true, but I suppose it could be mostly automated by going FinTS.
Direct billing works here too, but I steadfastly refuse to sign up for it, because I would rather decide when it is convenient for me to have the money leave my account. I will never forget that one time back in the '90s when all the utilities decided to withdraw a certain month's payments before the relevant paycheck had actually arrived! Not a good experience. Doing it manually makes it much easier to determine how much money I actually have to spend in my debit account at any moment.
I feel the same way, though some kind of configurable rules to determine when the debit comes out would go a long way to solving this. You can imagine an autopay interface where you say when your paydays are (every other friday, nearest-business-day-to-3-days-after-the-1st-and-the-15th [this is really my spouse's pay schedule], monthly on the 12th, whatever) and then rules like "always take the money the day after the first paycheck in a month, and only if it's less than $500". Bonus points if this gives you extra grace period compared to paying manually.