Battery power on common battery phones (in a common battery phone there isn't actually a battery, the terms here are mostly historic) is required for a carbon transmitter to work and, for DTMF dialing, it's used by some DTMF keypads. But battery isn't actually that important to most modern phones, which use ADC/DAC instead of an analog audio setup. Modems are similar, they have their own audio electronics that rely on their external power source. That's a good practice in general because battery power on phone lines is pretty variable and not always very clean, so you don't want to run ICs off of it if you can avoid it. The current is also too limited to use it for much, even indicator lights will run into certification problems.
The issue with the USB modems might actually be a bit different... more frequent use of single-pair digital business telephone systems in the '90s lead to problems with people connecting them to conventional modems. Some of these digital systems used unusually high battery voltages (sort of a PoE-esque method of providing meaningful current to the desk phones for their lights and displays), and there was a concern that they could damage the ADC on modems. So a lot of newer modems actually check the voltage on the line before they connect the audio electronics to it, as a safety precaution. There's an AT response that indicates that the modem is unhappy with the line voltage and I tenuously remember that at least some modems have an AT command to defeat this protection and make them connect anyway. Digital Line Detect is what the feature is usually called, but you might more generally call it a line voltage check.