I also have a NAS; a WD DL2100. I am connected exactly how you are, although the NAS is connected to a gigabit network switch, not the gigabit router which is across the house. I have an Asus dual band router and the Echo (as well as the Dot and the Tap) all connect to the 2.4G wireless signal (5G is overkill for Echo audio.) BTW, are you sure Echo has a good strong signal to router? (I do.)
All components work fine, so I really don't know what the problem is other than different equipment. I suspect either the router or NAS, Maybe connect NAS to a network switch into router?