There is a lot of confusion regarding what part of the OSI model is in use here. Let me see if I can help:
Remember that the OSI model is just a model. It doesn't represent anything in actual use. The TCP/IP model is a better fit for protocols in use on the Internet.
The statement "routers are just depicted as having and using the networking protocols and IP" is an oversimplification -- and that's where some of the confusion comes from. Router software has the full stack in order to run routing protocols like BGP as well as management functions (telnet, SSH, snmp, etc).
Is BGP a network protocol or an application protocol? The BGP process that runs on a router talks to BGP processes on other routers. BGP makes use of TCP/IP to facilitate that communication. It establishes sessions between peers and has its own messaging format and syntax. In that sense, BGP is an application that runs on a router.
BGP's purpose is to populate the forwarding table of the router. When the router makes a forwarding decision for an IP packet, it looks in the table for the next hop address, adds the layer 2 header, and transmits it out an interface. That process only involves layers 1-3. So if you're talking about how routers route, that's all you need to discuss -- and that's probably where the confusion comes from.
In summary, the forwarding of packets involves layers 1-3. the information used to forward packets comes from many sources -- one of which could be the BGP application running on the router.