Any other material for the crucible except quartz would introduce impurities from whatever substance it is made.
Quartz is already silicon dioxide. It could introduce only silicon atoms, which do not matter at all, or oxygen atoms, which also do not matter much.
The only alternatives to quartz would be other silicon compounds that have a melting point higher than pure silicon, e.g. silicon nitride, but all of them are more expensive than quartz.
In the past, besides the crucibles for holding melted silicon, quartz was widely used for the walls of the ovens (typically having the form of a quartz tube) where silicon wafers were heated for various operations, e.g. for diffusion, and also for the wafer carriers on which the wafers were placed while inside the ovens (a.k.a. "boats").
These uses expose the material to lower temperatures than in the crucibles, because the wafers are solid, not melted.
Quartz is still used for these purposes, but in many cases it has been replaced with polycrystalline pure silicon, which can resist to these lower temperatures and it also cannot introduce any impurities. In comparison with quartz, polysilicon has the advantages of matched thermal expansion with the silicon wafers and of higher mechanical strength at high temperatures.
https://www.mmtc.co.jp/en/products/quartz.html
"Semiconductor devices, the foundation of the semiconductor industry, are manufactured by creating a fine electronic circuit on a silicon wafer made of single-crystal silicon.
Single-crystal silicon is made from high-purity polycrystalline silicon (polysilicon) and is grown by filling a quartz crucible with the silicon before heating and melting it in the extracting machine.
A high-purity quartz crucible is the optimal container to ensure the purity of molten silicon when manufacturing single-crystal silicon."
Maybe read the thread you're replying to?