Not entirely, but your last sentence is true. Some issuers do PIN credit cards. I have one from First Tech Federal Credit Union that uses PIN and touts it as a benefit. When I used it in Europe, it worked exactly as expected and prompted for my PIN just like in the States. There are a handful of smaller issuers that also do PIN primary (mostly credit unions but at least one Florida bank with a card catering to Cuban trade does) and a few more that have PINs but the PIN is secondary so it isn't asked for unless the terminal's configuration insists on it.
Frankly, I'm baffles by these claims that US cards don't work with chip and pin because so far, if I use the chip, I have been required to use the pin.
I also have a corporate amex with a pin set up, but I've never used it.
Not true - BarclayCard Arrival/Arrival+ do. However, they prioritize signature first. They can be used at automated kiosks that require PINs, though.
That is for debit, which is an entirely different thing than credit.
There are no US banks that implement Chip & PIN. Heck, it's nearly impossible to get a Chip & PIN capable card for travel if you have a US issuing bank.
Swipe support will in theory eventually be disabled (e.g. authorizations declined by your issuing bank) at some point in the future when Chip & Signature is being rolled out.
If you're typing your PIN code into a terminal in the US, you are using debit though.