Are the Paradigm Studio 60's the .v5 or an earlier version? I assume the Rotel RC-971 is not a MK II?

I like the sound of NAD's C375BEE in its price range, and I'm using the same power amp section (C275). While the NAD C375BEE doesn't sound particularly lean, it doesn't have the same (excessive?) warmth  that some feel NAD is known for. It would be a slight step up in power over your Rotel, but I don't know how much difference it would make to the sound of your speakers. The NAD's strongest selling point is power, and you might be able to get more refinement in some of the competing products in that price range, but with a little less power.

Upgrading your speakers might pay off, but it kind of depends on how old they are, how much you're willing to spend for new ones, and if you have any real dissatisfaction with them.

You might be looking at a considerable investment to get a less harsh, and more natural sounding system. I'm not sure if replacing one component is going to do the trick.

When a system is categorized as "harsh and in your face" it's almost always the speakers that are the culprit. Perhaps the sound you liked and wanted 20 years ago is not what you want to hear now. FWIW Paradigm speakers from that era have always sounded slightly harsh and in your face to me. That's why I've never liked them. They've always sounded overly "Hi-Fi'ish" to me, not as bad as JBL L100's but in the same ball park.

Paradigm speakers of that era got a bad rap for that.  Much like the mentioned JBL series, they had a tendency to be placed in the price range of their associated components, or in mid-fi home theater set ups.  Part of the problem with putting uber revealing tweeters on entry level speakers is they show off the shortcomings of $300 cd/dvd players and $500 home theater amps and the associated $50 monster interconnects likely to have been used because the dealer cost on a $50 Monster cable is about the same as the $30 Acoustic Research cable that is not only a better cable, but better made to boot, but I digress to the evils of Best Buy.  

So, if this is the direction you are headed, I usually recommend only replacing your speakers when you have worked yourself upstream and you are really really confident they are the week point.  First of all, they commonly held misconception that speakers are the most distortion laden components we use is widely misguided.  Most main stream hi-fi speakers made in the last 20 years or so are a bit different here and there, but mostly neutral and pretty revealing.  And in my experience from hearing them, you often get way off course when you get in to the "ultra-fi" end of speakers, where they tend to do one or two things very well, but fall apart when asked to move away from that.  When a maker is building speakers for a man who makes $500k a year a maker can afford to make a piece of art, visually and sonically, because he doesn't have to market his product to very many people.  He is building his vision for his customers.  And because people buying at that price point are as likely buying it for arts sake anyway, which is perfectly fine, btw.

I guess that's a long way of saying don't underestimate the value of pouring your expendible cash into your sources and amps.  If you've bought speakers in the last 20 years or bought nice used speakers of the era, from a mainstream manufacturer, JMLab, Klipsch, JBL, Infinity, B&W, etc, they will absolutely NOT sound the same, but they WILL probably be basically neutral, and you are probably better off sonically focusing on properly driving each's power requirements and shaping any step from neutrallity with the signature of the amplifier/cable combination.

Bass traps in the four corners, and broadband absorption at the first reflection points, sidewalls and ceiling, and front and back walls. Then maybe a carpet of area rug on the floor. Could be enough to tame the harshness you hear. If not, it can only make whatever you get next better.