Beth, Ag. Giorgios is not that crowded in June ... the photos just make it seem so .... It's not "urban" whatever that means ... only little teensy groceries, and tavernas on the sand. Around the corner from my tiny place (Studios Naxos), the little grocery has eggs on the counter, in a basket. A farm lady drops them off every day. Urban??
In May-June it's very quiet at night, just the cafes and 3-4 tavernas for dinner ... yet in 15 minutes you can stroll along the seafront into the port town for all its attractions. You tell us you have a party of 6 ... to get them to agree on what to do together is like herding cats. But staying in Ag. Giorgios, each can do his/her thing .... go windsurfing, lie on beach, go into town, shopping in the maze of lanes or up to the Kastello at the tippy-top. No coordination necessary, so you can relax for once.
As for favorite Swimming ... I like a quick dip in Ag. Giorgios at breakfast time .... you have to wade out about 150 yards to where it drops off, but it's SO convenient, and there are trees right up to the sand. When I want more of a "Beach day" I take the bus out to Plaka ... a bit windier, in June very sparsely populated ... rent a beach chair & sunbrella for about €7, swim read etc, I take it to the 3rd Plaka Bus Stop, by nice Aronis taverna with deep cool roofed terrace, for lunch... if you order beverages from them, i think their beach chairs are free. Only beach I'd avoid is Ag. Anna -- very narrow & even in June hardly a place to lay a towel, a line-up of cafes right on sand that blare music.
The thing about the outer beaches is that they're sorta strings of housing along the shore, and Ag. Giorgios is more established. Those 3-4 outer beach places (for staying) were built right at the edge of fields ... thus they are mainly treeless and dusty, greenery is mainly bushes & stands of bamboo (and brave patchy lawn). Ag. G seems greener & shadier by comparison. The sea of course is lovely... and the bus will get you there.