It's not quite that bad. The thing you need on one end is a digital modem. You can buy a card containing a bank of these for the Cisco ISR G2 platform. The ebay cost is about $60. Then in the same router, you can put a T1 card, which the digital modems will use to accept calls (costs about $15). Finally you can buy an analog modem card (costs about $20) to accept calls from another modem. All of this would be contained in 1 or 2 rack units.
Really appreciate the pointers. Sadly as none of the modems in the video you linked sounded right either, I might be about to go down a hardware buying rabbithole.
I had this same problem a few years ago and went down the rabbit hole. I was looking for the US Robotics X2 and K56Flex handshake sounds and eventually found them.
Turns out I've been trying to track down the USRobotics Sportster voice modem sound from your first link. Even better as I can't find my 56k USRobotics, only the 33.6k one, so probably wouldn't have ever found that sound with hardware either.
Thanks; that's been lodged in my head for a couple of decades.
There are other methods but that's the one I would use. To see one of the other methods, try here: https://www.youtube.com/watch?v=yQ_hWS6ked8