Well Synology NAS is so much more than just a connected harddrive through the network... Also I don't think the router can handle NZbget, sickbeard etc.
I really dont know what software is available from the NAS manufacturers or the community. All i can say is that my Netgear WNDR3700v2 can handle transmission, smb, ssh, afp, an apache server and tvheadend pretty well on OpenWRT and there are many more packages (including nzbget) available.
I think you'll find a really big bottleneck if you're using NZB (with NZget or SabNZB). Even my NAS is too slow to give me the maximum speed of my fiber optic internet connection :( (stuck at 9MB/s)