Superconductors have a critical current density (Ampere/m^2) that varies with temperature and external magnetic field[0]. So if you want more current, you need to use a bigger wire (and/or make it cooler). YBCO HTS tapes have enough current density for power transmission[1].

[0] https://en.wikipedia.org/wiki/Yttrium_barium_copper_oxide#/m...

[1] https://www.amsc.com/comed-and-amsc-announce-successful-inte...