Under the carpet you'll almost certainly find particle board, OSB, or fiberboard rough treads. Those are probably bonded well to the stringers with construction adhesive, so some careful demolition is in order.
Then, flush joints are a problem in finish carpentry. If you were to try and retrofit wood treads, not only will the flat portion pose a challenge, but the nosings are unlikely to match well. Even if they have the same radius, slight vertical misalignment in the machining means the soft corners won't align. You'll be doing sanding and refinishing of finished woodwork to smooth the parts together, which is a royal pain.
After all that, the outcome will be just OK due to the break in the wood grain, stain color variation, and obvious butt joints.
You have two options, as I see it:
Remove everything and rebuild. This is as big-n-hairy a job as you imagine. I've done multiple such stair sets, and the number of moving parts is a bit mind-boggling. It's a multi-day job for even an expert carpenter, and a miscalculation in a baluster position or a skirt miter means starting over, with everything having been thoroughly glued together already. If you go this route, select your carpenter very carefully.
Just replace the plush carpet with something less conventional, like a good commercial carpet. Use something elegant and classic which complements the rest of your home's design. Think outside the box. Something that at first seems stuffy and pretentious may actually be a real winner. I've seen it done many times. Find a creative carpet salesperson and keep your mind open. Inspiration
With this approach you may want to extend the rough treads a bit. There's a lot of the inner face of the hardwood caps showing now, and a thinner carpet will exacerbate that, possibly to the point that the back edge shows. You may want to trim off the bullnose and securely mount new longer ones to improve final fitment.
By the way, wooden stairs are notoriously slippery to socked feet. A sleepy resident can easily end up in an unsustainable flight attitude*. That's the primary reason most are carpeted or have runners installed. I've seen wooden staircases in new homes retrofit with runners in short order due to this fact.
* One of the worst moments of my life was when I received a voicemail from my young, pregnant wife while on a jobsite out of town. She had just slipped down the last few steps of a wooden staircase I had built into our home and sounded terrified. Rarely have I felt so helpless and inept as a husband and future father.
