All EMV terminals allow C+P cards, even if some can be configured to not allow C+S cards (in the US that doesn't happen often, but can happen in Europe). What you referred to as your understanding from TSE is about what is being issued in the US, but it doesn't affect the C+P cards used for payments in the US.
Why would you want a C+S card if you have a C+P card? The latter are much more secure. The US banks don't usually issue C+P cards (see below, but it is starting to change now), but they're accepted just as well. In fact, even for C+S cards, signature is now rarely actually required.
Also, contact-less payments became quite popular in the US and from my experience contact-less cards, or e-wallets like Google Pay, Apple Pay, and such, are widely accepted.
Magnetic stripes are rarely used nowadays as merchants that still prefer them over EMV will now be holding the bag for fraudulent transactions. They're only used as a backup for malfunctioning EMV transaction.
American banks preferred C+S to C+P because, per The Atlantic:
The reason banks say they don’t want to issue PINs is that they’re worried it will add too much friction to transactions and make life difficult for their customers
This article is a bit old, published in 2016 during the EMV push in the US. As I mentioned, the things have changed slightly since. Some smaller banks or those that cater to high earner travelers issued C+P cards from the start, but larger banks are now starting doing it too (e.g.: Discover, Chase, Amex, and others will issue a PIN for you per request).