Thanks for the chapter!
Eliza is really too stubborn. They dueled and she lost to his superior magic, so rather than trying to maintain her hubris she should get him to teach her since it's obvious he's more talented and informed about magic. Also, forbidden spells aren't usually called forbidden just to make them sound cool but because they're either too powerful or too harmful to the caster as well to use. Using one in an assessment duel that she already lost is the height of stupidity and arrogance. Even if by some miracle she does finally beat him what possible gain could she get to make it worth it, especially after paying whatever cost makes that spell forbidden.
@Tensei21 Considering people were scared of Amber eyes and his unsocial behavior for someone to be OP and get familiar with him it's likely a member of the hero party he was with that defeated the Demon King before he reincarnated. As for her family being in trouble it's likely due to people having short memories, especially after 200 years, and as the MC mentioned people stubbornly clinging to the notion of peace while hiding and denying things that could threaten it so her family was in ill favor by both the common folk and nobility from the unpopular notion of wanting to be prepared when the peace ended.