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Why do the diamonds need to be transparent?

People interested in high-pressure chemistry follow what's happening with spectroscopy, UV/Vis or infrared. Fig. 3 on p. 376 has the Raman spectrum.

So they can probe/measure the contents with light, usually in laser form.

Contents are completely enclosed by diamond to hold the pressure. No way to probe otherwise as far as i know.

X-rays are also widely used to probe high pressure matter within diamond anvil cells

Here is an example at an ESRF beamline https://www.esrf.eu/home/UsersAndScience/Experiments/MEx/ID1...

Don't forget neutrons! Not as quick to measure, and not so good with very small samples, but well-suited to combinations of extreme environments beyond just pressure, such as temperature, magnetic field, voltage gradient etc. https://www.isis.stfc.ac.uk/Pages/Pearl.aspx

Amusing to see for an x-ray crystallographer that the neutron scattering coefficient for tungsten carbide is actually lower than for pure carbon. Neutrons are weird.

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