What are LoongArch's technical advantages over RISC-V? In other words, why should a company develop their own architecture (which then they need to push support for) rather than use an existing, free one?

I don't know about advantages, but lead times in the chip business are long and you're not turning around on a dime without very pressing reasons. Loongson has probably had many things in the pipeline as RISC-V started gaining steam. Their current processors are more advanced designs than the best known RISC-Vs.

Back when LoongArch was announced, RISC-V did not yet have enough (ratified) extensions to achieve feature-parity.

Even if it had, LoongArch is much more similar to MIPS. LoongSon would have had to make more microarchitectural changes before being able to tape out their first non-MIPS CPU.