Leetcode-style interviews became popular in the mid 00s, primarily because they were used by hot tech companies of the time. The thing to understand is that at that time, the idea of asking people to write code during an interview what sort of revolutionary. Prior to that interviews had little structure. It wasn't unusual for the hiring manager to make the decision, some times based on credentials, recommendations, or trivial-like questions.
This type of interview became wildly popular because it allowed budding unicorns to hire programmers of high quality at scale. The process was less biased than the alternative, reproducible and scalable. Here you have two blog posts [1][2] that show the line of thought at the time.
The reality is that big tech has elevated leetcode type interview to an art. They have reached a near local optimal through years of experiments and refinements. It is working well for them so they don't have the need to take big risks such as completely revamping their hiring process.
I love the topic of hiring and interviewing and I'd love to truly get at the bottom of which method works best. I like this article because it explicitly calls out shortcomings with typical alternatives that are not usually mentioned. I hope in the future a new crop of unicorns can take these practices to the next level and do a fair comparison.
[1] https://www.joelonsoftware.com/2006/10/25/the-guerrilla-guid... [2] https://sites.google.com/site/steveyegge2/five-essential-pho...