So it's literally impossible to prove a negative. I can't say for sure that the Buddha wasn't, and that the Shakya weren't related to the Scythians. In fact, I have a lot more sympathy for this viewpoint than a lot of other Tibetologists might. That said, while Beckwith is, as far as I can tell, as good at reading Tibetan sources as any other, and his book The Tibetan Empire in Central Asia is considered the book on Imperial Tibetan history and its interactions with China and the Caliphate, his later book Empires of the Silk Road is not seen in the same light.

Of course, I recommend you look into both books on your own, but I think the differences between them is pretty clear. Tibetan Empire deals with the nitty gritty of diplomacy, directly references primary sources, and goes into the details of how the Empires interacted with as precise dates as possible. Empires of the Silk Road zooms through centuries of history per page, pans entire empires, and culminates in what I can only describe as an angry screed against the evils of Modernism and the very concept of Post-Modernism.

I mention this because I think it's important to clarify Beckwith's perspective - and I use this term literally - as a scholar. While I did enjoy Empires of the Silk Road, and Beckwith has certainly done a lot of reading... I have so many questions about the book. His thesis seems to be that there was a "Central Asian Civilization" revolving around the Silk Road itself, which was not a trade route, but an axis of civilization. And that there are similar threads in the history of Rome, China, Tibet, Mongolia, Sogdia, Persia, India, Japan, etc. He compares, for example, the warrior cultures of Japanese Samurai, Roman Centurions, Korean Hwarang, and passages in Beowulf and the Mahabharata.

In a theory that I'm still not sure if it's anti-semetic or representative of something... actually there, he even describes a world-wide conspiracy of Sogdian and Jewish interests that sought to overthrow the Carolingian, Ummayad, Yarlung, and Tang Dynasties in favor of compliant regimes, with mixed success. The Carolingian Empire collapsed around that time, the Ummayads collapsed in favor of the Abbasids, and the Tang Dynasty descended into the chaos and madness of the An Lushan Rebellion (which was led by the likely half-Sogdian An Lushan ("An" being a name commonly associated with Sogdians at this time)). While the An Lushan Rebellion is quite famous and well known, and I trust Beckwith's knowledge of Tibetan and Tibetan history, especially since he's known as the authority on that subject (though if I was going to look into it myself, I'd want to verify with primary sources regarding the influence of Sogdians in Tibetan history myself) this is the first time I've heard of Sogdian influence promoting the Islamic schism into Sunni and Shia, or of Jewish influence in the Treaty of Verdun and the subsequent collapse and division of the Carolingian Empire.

And again, while I don't want to engage in ad hominem arguments, it does seem to feed into an anti-semetic conspiracy theory view of history... and Beckwith presents this continent-spanning conspiracy to overthrow four major empires as fact, with naught but circumstantial evidence... at best.

So were the Shakya and Scythians related? Or even interchangeable? Maaaybe. Like most of Beckwith's later book Empires of the Silk Road, I've only ever seen circumstantial evidence to point to this conclusion, i.e. linguistic similarities. And frankly, until we get that time machine to go investigate for ourselves, that's probably all we'll ever have. And based on my knowledge of Beckwith's work, that's probably what he's going for: that the Scythians/Shakya belonged to the same tribal group that was spread from the eastern edge of the Roman Emprie to Northern India. It's not a completely ludicrous idea. The Huns and Mongols (to name just two) would be spread along a similar geographic zone in later centuries and there's something to be said for how united their political organization was. That said, Beckwith probably wouldn't even go that far, but point to a broader linguistic and cultural similarity between the Scythians and Shakya, which again, I don't find super far-fetched, but is a lot easier to swallow than the vast scope of the thesis of the book from where this idea is probably coming.