The way the European system works is that there is one possible operation; the owner of an account can move money from their account to any other account.
If you want me to pay you rent, you provide me with a standard format bill, which includes your account number, the amount, the date to pay by, and an optional reference number (for use of the recipient to tell what I was paying for). To pay, I input those into my banking app (or more likely, read the barcode on the bill with my phone), verify that everything is correct, and press a button to transfer funds.
The point is, I don't have to trust you to do any of the banking right. I have paper trail from you in form of the bill you provided (which confirms that the numbers I used are correct), and from my bank that the transfer was initiated on a given day at a given time. There cannot be any error that I didn't make.
This isn't true at all. The direct debit system allows any participant to withdraw money from your account, providing just your account number and a claim that you authorised them to do that. You never have to directly tell the bank anything, and in many cases you can't even opt out of the process or deny access to an individual withdrawer through the bank.
The banks are meant to limit access to the direct debit program to companies they trust, and cut off access to bad actors, and make whole customers who were debited in error or by fraud. How enthusiastically they do this depends on the bank and the local regulator.
So you setup something like "account 123 can debit up to 100 EUR monthly" and the bank enforces that.
You could theorically withdraw money from an account by faking the procuration, providing you know the id details of the account owner, but you are still required to present your id card which contains a digital photo of you to prove you are who you pretent to be.
Fake id are maybe common in the us thanks to your broken SSN system but in europe, digital id cards are centralised from the moment you're registered as human being at your birth and banks have access to that system when they check your id.
Any attempts to withdraw money from your account leave a trace, and in case of fraud, you would be quickly with the cops on your trail.
The UK and Denmark don't have identity cards. At least in those countries, there is occasional fraud. The banks will fix it (at least in the case you describe), but it can happen.
Even ten years ago in Australia all of that was done through internet banking, and if I wanted to I could set it up to pay those automatically when the next bill came in (or not, it's up to me).
I've never written a check in my life (I'm 37), and whenever I get one I laugh and am shocked people still use them.
And to top it all off, the American bank cheque system is so insecure.
Functionally it's the same thing, except that instead of writing on a piece of paper that you're giving the bank authority to withdraw your funds, you're doing it online. It takes less time, it's more reliable, and you get even better record keeping, as you can enter notes when you make the transfer.
For me this week: Paid for a city license, paid the cat sitter, paid for a small magazine subscription, made a charitable donation, and paid the rent.
The cat sitter and the charity were check only. No other option. The magazine has a PayPal option, but I don't trust PayPal. The city license could have been done with a card online, but I'm not interested in setting up yet another account with yet another third party to mishandle my information. Paying the rent by check is free, but paying with a credit card is a $74 fee. And this is with a very large company that almost everyone knows.
People do abandon checks when possible. But for the most part it's the businesses that put up barriers. Or, in my case, a lack of trust, which has caused me to use checks more. I like the paper trail. I check my bank statement every month and at least once a year there is an error.
I am coming around somewhat on the trust thing. If I can use a virtual credit card number, I will. But not if I have to make yet another account with yet another password and yet more gathering of information that is nobody's business.
Edit: I forgot that I also had to send a check to a county recorder out if state to file some legal documents. But that was a cashier's check, not a bank draft. Still, the only form of payment accepted by that county government is cash if you live there, or a cashier's check if you're out of state.
Also, my accountant only takes cash and checks. I consider it a virtue since it indicates that she watches every penny. But I didn't put that payment in the list because it was a couple of months ago.
When your life is simple, banking is simple. As you get older and your life becomes more complicated, banking gets more complacated.