Cannondale used to make these shocks for mountain bikes and they were pretty common when I rode as a kid. It seems like they have completely phased them out of production. What caused this? Is there something inferior about this technology?
Cannondale still makes them. Under the form you present, they are only found in urban E-bikes. On MTBs, they evolved into a different form.
One drawback of this design is that the travel needs to be shorter than the head tube length. But MTBs evolved towards longer travel and needed to shorten the head tube to have good geometries. So Cannondale moved the mechanism out of the head tube and created a single leg design now known as the "Lefty" fork (mostly found on their high-range products).
More info here
It is a little difficult to give an answer that distinguishes between the pros and cons of the concept and any untapped potential it might have versus Cannondale's implementation of it.
With its very precise array of needle bearings on a non-round telescoping section, the Cannondale design is successful at being smooth and slop-free, but it's way too expensive and maintenance-intensive for its own good and the benefits it provides. Problems never really went away with how susceptible the design is to tears in the boot, which has a critical role to play in keeping contamination out and lubrication in. If the boot becomes compromised, things go south very quickly and a rebuild is often needed, to an extent where normal riders simply need more margin for error. The tight tolerances that keep Headshock largely free of perceptible slop also make it very easily affected by contamination. Headshock mostly exists/existed to be cute and tell a good visual story on the sales floor. On a practical basis its combination of low travel, marginal damping control, and high maintenance needs were too little benefit at too much cost compared to a conventional fork for mountain bikes and to an appropriately sized and pressurized pneumatic tire for utilitarian/road-going ones.
I won't speak too much to what's good and bad about the design in more abstracted terms. The Cannondale design sacrificed a lot to be very smooth and very free of slop when everything was working right, and also had a lot of proprietary tool, component, and support access hoops to jump through. It's possible to imagine someone taking the concept in a different direction, but difficult to imagine the result being clearly better than just using the right width tire at the right pressure. Note that the period Headshock arose in was basically the apex point for popular tire tire sizes being too narrow for almost every application in cycling, and of the dogma that higher tire pressure is usually better.
Suspension fork designs with one telescoping element like Headshock and Lefty are unaffected by increased stiction when the front wheel is side-loaded, as in cornering. That this is true to some extent compared to a conventional fork is provable and undeniable, and Cannondale loves to make a big deal about it, but the importance of it is highly debatable since loads on a bike track largely in-line while cornering, a fact you can read about in various technical work on bikes but that's easy to work around in marketing material because it's not completely obvious. Going over technical terrain at speed (mountain bike racing) it's not too much of a contrivance to say this effect provides some advantage, but it's a stretch on utilitarian or on-road bikes.
Other answers have covered some technical pros and cons. I will add the original Cannondale company was forced into bankruptcy about 20 years ago and broken into pieces auction off to various buyers, whoever bought the name and bicycle intellectual property started new design and production halfway around the world with completetely different motivations from the original company. So your question is sort of asking why Joe stopped making(or did not restart) Bob's old product.
(The bankruptcy was a disappointment. They had just released a new line of motorcycles[R&D was the reason for taking loans], and they were getting great reviews but they hit the market a few months too late to improve cash flow in time to pay the bills and the creditors were rather inflexible.)