Nah, this is revisionist history. Not only is the writing of Blizzard today demonstrably alot worse then back then, there are two things you need to consider: For one, the writing of their games was very good for the time that they came out in, very few games had better presented and more coherent and involved stories back then.
However, more importantly: Yes, the writing of Blizzard was not high-tier art and you mostly got the broad strokes of what was happening, but for the games that Blizzard was making back in the day the writing was extremly effective. The writing had engaging overarching narratives, easily memorable characters that you could get invested in and it did an excellent job at building the world. Yes, on a micro-level the writing is of course not up to the level of what one might expect from movies or books, but it did an extremly good job of selling the idea and world behind the games.