I have a 5 node mesh network strewn around the office, mesh connectivity is dandy, I have RPi servers at each node, name resolution on the mesh network is functional, all is well in the mesh.

I have one of the nodes connected to the office network, and I can get to the Internet from the node and from any of the LAN ports on that node. The node is set to "mesh gateway" via the checkbox. 

The other nodes have no route to the Internet. Do I need static routes on the other 4 nodes? What purpose does the mesh gateway checkbox actually serve?

Thanks in advance!

When the checkbox is turned on the router will try and verify internet connections with pings.  If the node receives a response it will publish routes to 0.0.0.0 (default route) in the same manner it publishes routes to networks reachable in direct mode (called HNA Announcements). In addition firewall rules are added allowing forwarding from the mesh interface to the WAN interface and the node will perform a NAT translation so everything appears to come from it's IP address on the WAN network.

the 0.0.0.0 route should show up on all other nodes instructing the node to forward packets addressed to anything not advertised as connected to the mesh network.

Wit this  you are giving every other mesh node access to your office network and the internet to do with as the mesh nodes and client please.  No filtering or restrictions are done by the meshgw nodes by default and if running as part 97 you need to be careful what runs across the network and read the part 97 rules to be sure anything across the network is compliant.

The 0.0.0.0/0 route should show up in the OLSR status screen as I recall (however I can't check on mine right now)  as an HNA annoucement.

No GUI exists to see the full routing table at this time.  You would need to log into the node and run the route command manually to see the actual routing table.

I'm more familiar with the Ubiquiti build than the Linksys build so I'm going to defer a lot of this to someone else for the exact specifics.

My guess is that your main node for whatever reason is not advertising the route.  You could look at the OLSR status screen and get some info about it.  Some information would have to be gleaned from the CLI.

A quick test would be to ping 8.8.8.8 from the command line and see if you get a response.  If not than that is why the node is not advertising the internet because it can't reach a test host.