By definition, frames (what you have is a broadcast MAC address which goes on an ethernet frame, not an IP packet which uses a broadcast IP address) do not cross a layer-3 boundary. A router will strip off a layer-2 frame before forwarding the layer-3 packet encapsulated in the layer-2 frame. The router will create a new layer-2 frame with which to encapsulate the layer-3 packet for the next hop. Layer-2 broadcasts are not forwarded across a layer-3 boundary; neither are layer-3 broadcasts, except in special cases.
MAC addresses are only significant on the layer-2 domain because they are only needed for the layer-2 frame. A router will have an ARP cache for each ethernet interface which it has, just as a PC would have an ARP cache for each ethernet interface which it has. Neither a PC nor a router is sure to have all the MAC addresses for every device in the layer-2 domain.
Routers are only needed when a host needs to communicate with a host on a different layer-3 network. Any host can communicate with any other host on the same layer-2 network (there are some corner cases like Private VLANs where this may not be true) without going through a router.