Why did we move away from this model of web development? If I remember back that far, it seems like fairly soon after this was when the meta starting telling us that frames were bad, and using was bad, etc. It all honestly seemed a lot easier than what has come after.
The direct answer to your question is that the idea of "responsive" web design gradually took over when smartphones became popular. People liked the idea of not needing to create a separate mobile version of their frontend, and instead simply adjusting the styling and structure of the page in response to the size of the viewport. Tables were bad at this because you couldn't use CSS to arbitrarily move around the way…
Table-based layout was already falling out of favour before this happened. Even when you had fixed sites that were predominantly desktop, divs and spans (coupled with CSS) made life a lot easier to put things where you wanted them and lay out a site faster. It was also cleaner when reading the markup.
From what I remember, the move away from tables happened around the same time external CSS took off and you could partition your site with reusable styling instead of framesets or tables.
Responsive design then sped up the process a few years later.