No post-processing tool in the world can replace lost data. That's basic physics. You can use filters and sharpening to create the illusion of a cleaner image to untrained human eyes but you can't replace actual image data.
The computer guys have an expression for this: GIGO. Garbage In, Garbage Out. With digital post processing, you can do a better job of concealing the garbage, but it essentially still is garbage.
That's why there are still tripods.
This isn't some new discovery either. Pretty much all of this was formally codified in optical physics terms in the nineteenth century.
The main benefit to digital cameras is that you can take 10+ images per second and just maybe one of those handheld shots is clearly that the rest. That was not cost effective 20+ years ago when a 36-exposure roll of slide film & the requisite E-6/C-41 processing totalled $20-30.
The pro 35mm SLRs from that era had mirror lockup (MLU) to reduce the image degration from the vibration caused when the reflex mirror flipped up. It is also why there were remote cable shutter release cords.
Even today, not all tripods are created equal. Tripods have different performance characteristics based on size, weight, construction material. A heavy Gitzo metal tripod with fewer leg segments is going to transmit less vibration than a flimsy lightweight carbon fiber tripod.
For those of you who say "I can get great handheld shots with my 600mm honker", I believe you. I'm also convinced that you deleted 50-100 crummy shots on your dSLR. In the old days, you would have burned through $100 of film/processing. Even if you could shoot at 10fps on the best SLRs, you'd still have to reload.
High-capacity digital storage cards have changed the game. In the old days, professional underwater photographers dived with 10-12 SLRs in waterproof housings because each SLR body was only useful for 36 exposures.
For the best final image quality, you need to start with good source material.
There's a reason why Olympic biathletes and other shooters use support. Even a tenth of a millimeter movement during a camera shot can result in a lot of blur particularly in telephoto images. Again, this topic has been beaten to death for decades upon decades.
What you say is certainly true. That said, I'd also say that it is easier to get better hand-held shots on modern longer lenses and modern bodies via image stabilization (IS) tech (in the lens, body or both). Higher ISO capabilities in the same modern bodies help too, though that degrades image quality by reducing dynamic range and SNR (which reduces color fidelity, among other things). Noise reduction software can help to some degree but to your point, we're already starting with a degraded image with less latitude in post.
When I'm ambitious (read: not being the lazy guy I normally am), I do carry my tank of a tripod and gimbal on my bird/animal shoots. It lets me get the best possible image quality by allowing me to have a better balance between being closer to base ISO and having a reasonable shutter speed than I can with IS technology.