I wouldn't trust them for anything important. They only showed that they couldn't do it, not that it couldn't be done. They had a few days, a small budget and a dozen or so helpers. Archimedes would have had a military budget, years to refine the idea, and countless helpers. For me the main interest in the episode was illustrating some of the problems to be overcome.

Anyway, it's kind of irrelevant here. The accounts of Archimedes shows that the principle of making fire by focusing light were well known back then.

It would be less effective, though. The trick with solar fire is to collect large amounts of light. Magnifying glasses have the same problem - I've never seen a wallet-sized one work in the UK.

Glass lenses are hard to make big, and also get heavy, but you can get flat plastic Fresnel lenses. You can get big ones which would probably be more packable than this reflector.