Starship designed by comity 2.0 ? Only this time around we go for the "kewl" factor and TOSs canon aside ? ;-)
The Dilithium processor thing - it's an engine part, essentially. Probably a really common, easy to come by one in normal circumstances, except that there's a war on and T'Kuvma's ship hadn't had any resupply or aid from the other Klingons in 6 months. The Shenzhou was an old, heavily battered, derelict ship. Presumably with an old warp drive. Even if Starfleet had gone through and salvaged what they could from the ship (don't recall any specific dialogue stating they had, but I have only watched the episode once), I doubt it would even register for them to take it. They probably didn't even know Voq needed that very part to get the ship moving again. It's the equivalent of soldiers driving a Jeep in a war zone, getting shot up by the enemy and leaving it behind. It's a knackered old Jeep, why would anyone go back and strip out the spark plugs in case an enemy vehicle breaks down nearby?