You are right, I left out an important part of my budgeting setup: Every month, after income, everything above a set amount is transferred automatically into a savings account. So at the beginning of each month, I have the same amount of money in my checking account.
Still, even just the number going down tells me exactly how much money I have spent this month – like looking into my wallet would. But I have the added benefit of knowing what I spent it on even long after the purchase.
I don't always review the paper trail, only if my spending is higher than what I expected it to be. But I always check the balance before any bigger purchase. In both cases, I know to spend less in advance and can make an informed decision.
* I am translating from German here using some quick Google searches. I hope these account types make sense for a US reader.
I believe you are conflating the goal of "spending less" (which is hard) with a simpler goal "knowing more about how I spent my money" (which is easy). You claim to review your paper trail to gain insights, but again, even if you truly did this consistently (and knowing human nature, I doubt that you do) this in no way means you spend less.