When I saw the title I was expecting a report that the system was a bit dull or overblown in the bass, not harsh. The A5's can easily overdo the bass and become flabby and boom but the mid range and treble is delightful and very smooth.
I have A5's in an all Naim system (NDX Streamer, 282/hicap/200). I initially used them with a 202/200 combination which isn't far off the Supernait in ability but has a reputation of being harder sounding than the Supernait.
With that combination and a variety of sources at no time have the Spendors sounded harsh, in fact one area for improvement was attack and bite, although a recent change in preamp to the 282 pretty much rectified that.
I think you need to look beyond the amp/speakers for an explanation. Trying a different source would be a good idea but I would also ensure that you are using Naim speaker cable which at the price is good value and you need to go a long way to better it with Naim equipment. Other things to look at are the positioning of the speakers and the room, is it a lively room and could some soft furnishings or a rug help?
Also, make sure you leave the amp on all the time, Naim gear needs to be left on to perform at its best.
Good luck
Richard