The biggest problem with this narrative is that scholars now believe that the stories of Qin Shihuang are greatly exaggerated and did not actually occur, and some of the things you mention above are misconceptions. Most of the accounts we have come from the Shiji written by Sima Qian, which is known to be biased since it was written to legitimise the Han Dynasty at the expense of the Qin Dynasty.

1. It is now believed that some element of book burning did occur, but certainly not to the extent that was described. Even within the Shiji, it was written that 2 copies of every book were kept within the Imperial Archives, though those were later destroyed by rebels. In reality, it appears the book burning campaign did not actually include historical or philosophical texts. The reason is simple, Qin Dynasty records indicate that the palace continued to employ renowned experts in the Shijing (classical poetry), Shujing (history), and an entire ministry of experts in the Yijing (rituals) in the Ministry of Rites. The largest evidence that Han Dynasty scholars made up the narrative of a Confucian purge is that the central claim revolves around the burning of the Confucian Five Classics, which would have been scandalous in Confucian Han society. Unfortunately, the Five Classics were not yet affiliated with Confucianism until the mid-Han Dynasty. On top of that, a treatise written by the last emperor of Qin cites heavily from the Spring and Autumn Annals, the history of the State of Lu written by Confucius himself. Much of the texts alleged to have been burnt survive intact to this day, while some of the others (especially the Mohist classics) were already long lost before the burning supposedly took place.

2. The implementation of Legalism in China greatly preceded Qin Shihuang, and rather than being his goal was what enabled his victory. Legalism was actually implemented by Shang Yang, who was the Chancellor of Qin to Qin Shihuang's great-grandfather. Shang Yang's reforms ended rent-seeking amongst the nobility, centralised and united a country on the constant brink of civil war, and enacted efficient tax and agricultural policy that enabled Qin to rapidly develop a military-industrial state surpassing its neighbours. On top of this, the reforms Shang Yang enacted were not new, but were based on those enacted by Wu Qi in Chu just 10 years before, albeit much more drastically. Throughout his reign, Qin Shihuang made little contribution to the study or practice of Legalism, in true Legalist tradition he only served as the executive, and delegated legislative efforts to experts and scholars like Li Si.

Finally, there is the issue of exactly what Legalism is. Pure-form Confucianism, Mohism and Legalism never actually existed, and would have been incredibly extremist and indeed bizarre to anyone. A pure-form Confucian would have believed that rituals are the purpose of life and nobody should do anything that hasn't already been ritualised. A pure-form Mohist would have believed that families, friends and countries are all false, we should all live in a massive structured homogenous community. A pure-form Legalist would have believed that the state is God and anyone who breaches any law should be executed. Obviously all of these are insane, and scholars knew that well, which is why the prevailing intellectual current was for rigorous debate, and a position that sat somewhere between these three great philosophical traditions, though today all three have essentially been merged into Confucianism.

3. It is highly improbable that Qin Shihuang was anti-intellectual, as he was himself a well-known scholar and deliberately surrounded himself with even more well-known scholars, like Li Si. The stories of him burying scholars alive are highly suspect because at no point in any text are we ever given the names of any of those scholars (Chinese historical texts are usually obsessed with naming everyone, much like the Bible).