No, it’s more like HomeModem ←A→ Exchange1 ←D→ Exchange2 ←A→ ISPModem. The digital parts were all inside the telco’s networks that connect the exchanges to each other.

> Couldn't you send data at megabits per seconds over a mile long copper wire without using modems at all (using just UARTs?).

No. The exchange is sampling the analog signal coming in over your phone line at 8kHz and 8 bits per sample. They just designed modems that sent digital data over that analog link, in a way that would line up exactly with the way the exchange will sample it.