Will a router set up with EIGRP route traffic from router not part of EIGRP

I’m having a friendly argument with a coworker. He thinks that the a remote site router isn’t communicating because it’s not part of EIGRP.

I informed him that if the gateway of last resort is set to the next hop router which does have routes and happens to be a part of an EIGRP AS, that it will route the traffic to a known route, this EIGRP is not needed on the site router.

My understanding is that EIGRP is for sharing routing tables not for actual routing itself. He insists that if traffic that’s not coming from a router configured with EIGRP it will be dropped.

Am I misunderstanding EIGRP?