We just don't have that much fraud instead.
One big difference is that in the U.S. cardholders are largely protected from credit card fraud (not debit card fraud), so the card vendors have to take the risk and so have robust anti-fraud measures (both before and after payment). Largely it is the merchants who have to prove that there was no fraud. Whereas in Europe the burden of evidence (not proof) is with the cardholder.
There's also a large difference between counties. In the Nordics its ubiquitous, I haven't carried or needed cash for almost 20 years. Meanwhile Germany has barely started to use cards.
In the US, you simply have no choice if you want to eat in a restaurant, so people are used to it. I'd expect total skimming rates to be higher in the US, since magnetic stripe transactions have been phased out in effectively all other countries. People don't care because they don't directly pay for the resulting fraud out of pocket. As a society, of course everybody still pays for it.
> Largely it is the merchants who have to prove that there was no fraud
No, in-store, it's the issuing bank that's liable, even in the US (unless the card is PIN-preferring, which is usually only true for foreign cards).
You get nervous about giving your card to a waiter because you’re in a foreign place with a nonsense payment system worst than most developing countries and it’s not something you’re ever asked to do anywhere else.
... No. It's just not done in Europe because it is _not permitted_. Chip and pin was made mandatory decades ago, and for that the cardholder must be present. It's not unreasonable to be disconcerted when something which just doesn't happen in your experience (the waiter takes your card away!) happens.
Skimming is also mostly now a US thing, as non-US cards no longer require a magnetic stripe at all (the only US has a stay of execution; Mastercard plans to completely phase out magstripes by 2033, and I think Visa is similar).
The US has _far_ more card fraud than the EU (about $15bn vs $1.5bn/year), largely for technological reasons (the US doesn't require, and many issuers don't even support, chip and pin, or 3dsecure, and of course there's the magstripe thing, though most card fraud is online these days where 3dsecure is more relevant).
> Whereas in Europe the burden of evidence (not proof) is with the cardholder.
This is the case for chip and pin transactions. It would generally _not_ be the case for foreign non-chip-and-pin transactions, though cardholders may not necessarily be aware of this.