"All types are not equal" isn't a mistake, it's a feature, and that's what makes types flavorful from a worldbuilding and in-universe perspective, as well as a gameplay design perspective.
Different types are intentionally not equal because they have different roles to play from a gameplay standpoint in addition to their flavor standpoint. Bug-type is a weak type because it's basically an introductory type, it's an early game type and its classical members are the likes of Caterpie and Weedle and their derivatives. Small insects who in many cases undergo the metamorphosis process from larva->pupa->imago very quickly thanks to low evolution levels and act as early game crutches, being relatively strong early on but quickly falling off to be replaced by better Pokemon later on so you make room for Pokemon introduced later in the game. Bugs are a fairly intuitive "early game" type of Pokemon as insects are commonplace in our world and the metamorphosis process is very intuitive to a kid, and they interact with the three starter types in intuitive ways (and Flying, another early type, has an advantage against it, because of the classic "birds eat bugs" thing). It's an ideal early type and is weak as a result of that.
Meanwhile types like Dragon, Steel, and Fairy are more powerful because they are later game types, and conceptually are less intuitive and more abstract, with Dragon covering a very abstract concept and gameplay wise resists all three starter types and is only weak to itself, Ice (a late type), and later on Fairy. You get the drill.
It not only makes it more satisfying to use different types as per their flavor, like winning with Bugs, but it makes any newly introduced Pokemon who deviates from the traditional mold a type has established for itself stand out that much more. Bug has traditionally been a weaker type with weak Pokemon, but then you have something like Volcarona, a very powerful late-game Pokemon who is a Bug-type. Volcarona is standout and memorable to people because it combines being a Bug-type with being an exceptional and unique late-game powerhouse. In general many of Gen 5's Bugs are great, Scolipede, Durant, Galvantula, etc. which helps make them all really stand out: because they're Bug-types that are powerful. And they have more standout designs to boot. Or in the case of Dragons, which are traditionally associated with legendaries and pseudos, then you get to something like Druddigon or the Applin line or Turtonator+Drampa. They're dragons, but statistically less impressive and very average Pokemon, and even in terms of design and lore they are comparatively unremarkable in every aspect. A Dragon-type that isn't particularly powerful or impressive stands out because they're so, well, boring and average, in a type filled with exotic and powerful designs.
The fact that types are of different strengths relative to each other, and the fact they are often used and distributed in-game in different ways, makes them flavorful and more interesting in that regard.