Are superconductors primarily for DC transmission?

[some] people believe...it would power a phone indefinitely without an additional power source. Is it safe to say anything it powers is a resistor and this cannot be our interest?

Correct. A superconductor can transmit power from point A to point B without any Ohmic loss, but it is not an infinite source of power. A superconducting magnet can act as a source of power, but only for a limited time. The magnet can store a finite amount of energy, but it does not create energy.

is our main interest for DC power distribution?

I have not heard of any real-world example of a superconducting power transmission or distribution system. Superconductivity only occurs at extremely cold temperatures, and the cost of keeping the transmission line chilled to such temperature would vastly outweigh the cost of energy that is "lost" in an ordinary transmission line.