Given the prevalence of damage against castles being done by units that cost food and gold, and the fact that gold can be acquired by trading, while stone has no similar way to obtain, should castles cost stone to repair?
No, if you have a castle you dont need stone but you need gold to win your game even you have a castle. So gold is more important then stone after you built castle. Also if you have trade: You can buy stone.
By the way i think, chemistry should give less addional damage against buildings to trebs but trebs damages should reduce repair speed of villagers; so, opponent cant repair his castle for 383746727272 years and keep his position.
Repairing does not a comman thing in team game, there will be some units to snipe your reparing vils when your castle get sieged (%99 game will be at imp and your ops can throw some units )
If you are playing 1v1. You can run dry on gold which %80 lose for you.