Can the store turn that off? I thought that was something the bank controls. Or maybe that was just for contactless?

I'm pretty sure with EMV that the processors don't care about signatures anymore, at which point it's the store choosing.

A little bit of reading I did suggests that the till software is often the stopper.

Signed transactions depend on the bank; it means the card only supports SEPA Debit, so the supermarket has to manually debit your account later, that's what you sign for.

Modern cards (anything that supports EC) can handle using only chip+pin or contactless.

It’s the US spec. EU side it’s chip and pin. Which arguable is much of a sameness since you’re touching the key pad

Signatures very much still exist in the EU; if the card you put into the reader only supports SEPA Debit and no immediate transfers, the shop has to do a manual SEPA Debit in the evening using your signature as authorization. A few banks refuse to upgrade to anything newer.

Here (Germany) it was recently announced that banks would raise the limit up to which no validation is needed for (maybe just contactless?) payments. And indeed since then I haven't had to confirm a transaction, which is why I asked.

It's an option. For example, Walmart doesn't require a signature on any transactions.