Hi there welcome and congratulations on the bike!
Yes the bike isn't perfect and as you already know the suspension does have some shortcomings. However for the price how good is it? After setting the bikes suspension sag for your weight and application I am sure you will enjoy the bike for what it's for. I also softened the front end and have a set up using all of its travel is a much better proposition to what my other bikes have required. Yes tyres make a big difference and I answered (or I tried ) your question on the "Tire & Misc Equipment " thread.
Good luck with it, compared to your other bikes may not be as fast but very stable and certainly more comfortable!
I owned a hypermotard when they first came out and loved it so, I've always like the looks and riding position of some of the sport naked bikes, the Z1000 was always one of my favorites and the GSXS1000 was another bike I've always liked. After commuting for a bit on my R1, I realized that I needed something comfortable and yet...decently fast (DC area commuting) and some decent handling. Hands down, the GSXS1000 checks all the boxes. The wind blast is not bad and after dialing in the suspension and tire pressure, the bike is exactly what I was after. Of course, it cannot handle like a supersport but as far as moderate riding and speed, IMO this bike is completely capable.
As you said, the forks stock were too stiff so, I softened the from compression/rebound and also the rear which now has balanced out the bike so it feels planted and turns in nicely.
My old commuter was a DR650 with supermoto rims on it so...this bike is worlds ahead however, for what the DR was, it did a good job. It was a bit lacking on highways however, on back roads it did well.
I really like the GSXS, it checks every box and after getting the suspension dialed in and figuring out air pressure with the stock tires, I am very happy with this bike!