It's (quite) a bit more complex (and expensive) than that. You can't just push a lump of carbon really hard into a sheet. You have to worry about all sorts of things to get a high-quality crystal, and making crystals that are both optically high-quality and large is very expensive.
What was solved 50 years ago? Just because we can make synthetic diamonds does not mean we can make synthetic diamonds of any arbitrary size or quality. In fact, it becomes exponentially more expensive in proportion to the volume of the produced diamond. Most artificial diamond companies limit mass to 2-300mg per diamond because it gets much more expensive after that. Of course, we are speaking of optical-quality monocrystaline diamonds here. Polycrystaline diamonds are less expensive, but unsuitable for optical equipment (e.g. cell phone screens).
That's not to say the problem is hopeless; in fact, there has been some very interesting research done in recent years on low-pressure microwave-based diamond synthesis that could allow for very large, kilocarat-scale diamonds. But for now, it is not feasible (as far as I know) to produce diamonds in the shape and size needed for cell phone screens, and certainly not at low cost.
If your idea of the cost to mine diamonds is inspired by its retail price, you would be surprised to discover that the retail price is totally artificial. The markup is astronomical. Also, diamonds are not at all rare.
Hmm, they offer discs up to 12cm [0] in diameter. However they are made using "the chemical vapor deposition technique allow[ing] the synthesis of diamond in the shape of extended disks or wafers. Under optimized growth conditions the properties of these disks approach those of perfect diamond single crystals."
I wonder if how the mechanical properties compare to single crystal diamonds.
Sapphire still has a very high hardness, much better than glass of any sort.