I don't know about everywhere. But in my country, when the terminal fails to read the chip, it asks you to swipe the magstripe. Then it realises "oh this card should have a chip" and forces you to insert again.
After about 3 cycles, it gives up and accepts the magstripe (plus the pin)
In Norway, on the 3rd cycle, if it actually accepts more tries, you have to sign the receipt and if the purchase is more than 1000 NOK/130ish USD (might be higher) they have to phone the bank to verify/get a verification code to put on the shops receipt.
Often it will just deny the transaction completely and the shop queue grows _looong_
My understanding from working in a bank in the UK is that it is the shop's discretion to use signature instead of PIN with Chip & Pin however the bank does not refund/guarantee fraudulant signature transactions like it would a PIN code.
I've lived for long durations in the UK, France and Switzerland I've never seen this happen in any of them. However, I have seen it happen twice while traveling for a few days in Norway.
Curious question: can you request a replacement card be issued? Broken chip sounds like something reasonable to replace for (insert mumbling about poor user experience and whatnot here).
I had this happen with American Express. Went to two stores in a row where the chip failed and I had to use a different card. Went online to request a new card and they sent one out immediately with no charge or really any questions (I think I did say that the chip stopped working, but that was it).
Later, before the new card arrived, I tried using the old one again and it worked, so I just chalked the whole thing up to those two stores likely using the same payment processor that was having issues that day or something.
Most U.S. banks implement Chip and Sign, though, and enable the Magstripe fallback because retailers still haven't certified their systems end-to-end for Chip and PIN.