In the video they are building the bricks on-site. I've done this before. And also built a kiln on-site with mud bricks (with a small team), though it takes a good design to get it to the high temperature required. Rammed earth is another natural building alternative where a formwork is built in-situ, infilled, and tamped down, once again with on-site materials. And from what I've seen it's absurdly strong, and can be polished such that it reflects like marble.

For extra challenge, any idea if a solar kiln could be practically used for mud bricks?

You need a temperature of about 1000°C to fire bricks, but more is better to get the moisture out. From cursory googling around it seems that a solar kiln wouldn't cut it, unless perhaps a very large solar collector was used... which upon further cursory googling yielded this:

https://helioscsp.com/concentrated-solar-power-for-high-temp...

https://cordis.europa.eu/article/id/421794-the-next-solar-re...

Interesting, thanks. :)