Why do American restaurants/bars require you to sign the receipt?

They do not require it. Even if you do not sign, you will still be charged. They would highly prefer you to sign it, so that there was proof you reviewed the charges and are quite literally “signing off” on them.

Not signing, though, won’t get you out of paying. All it means is that the business can charge you exactly what the total was. That is, no additional tip, etc.

if there is a later dispute about a charge then the signature offers proof that you reviewed and approved the charges

In the UK over a certain amount you have to type your PIN into the machine, which is effectively signing the receipt.

The credit card companies don’t require them, but individual merchants still can. at their own discretion.

From what I’m seeing now, the ones holding onto the practice most strongly are restaurants and bars, and that might be the key to this.

I read once that tipping is different in Europe, that it’s included right on the bill instead of being added by the customer?

Because what I’m seeing says that one reason bars and restaurants might want you to sign the bill is that if you add a tip to it, you’ve changed the amount and they want you to authorize that change.

So if that’s not a factor in Europe, it could explain the difference.

The US uses “Chip and signature” while the rest of the world uses “chip and PIN”.

An artifact of the original US desire to prevent any big bank from getting too much control, the decentralized banking system has slowed down access to things that the rest of the world takes for granted.