Can you post your export and also explain what the Cisco is doing.
A 2960 is a layer2 device so no routing there.
As it seems the non-working systems are residing in a different network segment (broadcast domain)
They need a default gateway inside their own segment.
If the diagram is correct then rextended has already pointed to a potential problem. e.g. According to the diagram the host with an address in the range 172.16.2.0/24 is supposed to use a gateway at address 172.16.1.91. That would be a problem because address 172.16.1.91 is not in the range 172.16.2.0/24 so the host has no means of reaching the gateway address.