This is because a setting in the WebSphere Integrated Solutions Console (fka Network Deployment Console) and not the node agent service startup setting controls the startup state of the application servers.
The restart state behavior of the application servers is determined by what IBM calls the Monitoring Policy settings. These settings tell the node agent what it should do with the java processes after both an internal (to WebSphere) abnormal shutdown and a complete node restart.
The settings we are concerned with are Automatic restart and Node restart state. The first controls how WebSphere will react to an internal application server failure and the default is set to True. The second is how WebSphere will handle a complete node start (reboot) and the default is set to Stopped. This means that if you restart the physical box (or it reboots itself) you will have to manually start the application servers or WebSphere cluster of application servers in the WebSphere Integrated Solutions Console.
My philosophy on automatic starts is that it should be disallowed only in situations where automatically restarting will either undoubtedly fail (ex: failing back an E1 enterprise server on a W2K3 cluster) or could cause corruption. Since WebSphere is mostly at the presentation layer and since Oracle is making extensive progress with Failed Transaction Recovery, I prefer that the application servers start automatically on node restart.
To have the application servers automatically restart, set the Node restart state to RUNNING in the Servers> Application Servers> server_name > Server Infrastructure > Java and Process Management > Monitoring Policy section of the WebSphere Integrated Solutions Console. Click OK, save your changes, and test.