The Mersenne-Twister approach should certainly not be studied anymore, even if some popular old libraries still use it. It fell long out of favor, is too slow, and not good enough.
Modern PRNG's can be tested with Dieharder, TestU01 or STS and benchmarked. This article only talks about primitive old LCG's (not any good one) or MT.
When you say Mersenne-Twister isn't good enough, what are the other shortcomings apart from speed? It seems that even modern versions of Python are continuing to use it...
The homepage might come across as a a little overzealous (for example ChaCha quality listed as good rather than excellent), but generally has good points.
For example, for one of his arguments, he specifically chose a generator called pcg32_once_insecure, which the PCG author does not recommend due to its invertible output function!
Personally, I have read both arguments in detail and I would always use PCG or even a truncated LCG over xoshiro, which has a large size in comparison, potentially worse statistical properties, and no gain- faster in some benchmarks and slower in others.
Yeah, xor is simpler than multiplication in terms of hardware complexity- luckily, we have the multiplication circuits built in, so may as well take advantage of them.
Note, however, that it has an extremely large period which is a need for some specific applications (but I do agree that, for most use cases, it should be avoided).
Modern PRNG's can be tested with Dieharder, TestU01 or STS and benchmarked. This article only talks about primitive old LCG's (not any good one) or MT.