That is literally how checking works in the united states. You can withdraw money from a bank account if you have a routing number and bank account number and "permission." The permission could be a signed check, it could be an auto-pay ACH you set up by handing a paper form to someone, it could be a web-form you signed without looking over the fine-print.
Most extractive companies (e.g., gyms, verizon) have so many fine print and so many lawyers that they can literally take anything from you without recourse. You'd spend tens of thousands in lawyer fees without getting the money back. You could go to the attorney general's office for your state. Good luck with that, unless its a common issue, you just have to eat the cost as part of being American.
Or it could be nothing. In this case, its still on you to figure it out and get the money back in time.
The gym industry is very predatory and it needs far more regulation.
It should be noted that in the US, some banks will allow you to request a default ACH block/lock on your account, and I highly encourage you to ask for such.
That said I haven't heard of companies or fraudsters abusing it
"A no-questions-asked, fast and simple Refund procedure available within eight weeks of the debit date."
Within 8 weeks of the charge, you just phone (or otherwise) your bank and ask them to undo the charge.
Maybe it still works that way in some parts of Europe, but in others it has been changed to work differently, or doesn't exist anymore and has been replaced by other systems, depending on how you want to put it.
What do you think a check is? It's a piece of paper with your account number, who to pay, and how much. There's a signature field, but it's not checked too closely (I don't think my bank even asked for a signature sample when opening my account) and signatures are trivial to forge.
My bank doesn't give me a cheque book any more - haven't see a person use a cheque in about a decade. I can't believe they'd let someone cash a random cheque with no authorisation.
Which is why the rest of the developed world wonders: So why don't you stop that and catch up to, say, the 1990s?
Our payment infrastructure ... has some legacy features.
It happens in Australia as, just fill up a direct debit form which has the client's name, account number and sign.