I think part of this is, again, you're using "stat block" really oddly.
I think that we are getting side-tracked in abstraction. Let's be concrete. It's my contention that fights with a pack of rats, a pack of wolves, a snake, a crocodile, a boar, and a brown bear should all be distinguished by more than simply how many hit points they have or how big their attacks are. Each should have some sort of signature features that makes fighting with that creature feel immersive, believable, and rich.
So the pack of rats should highlight that the rats are small and relatively hard to hit, and that they have some sort of swarming attack where they climb foes and try to use the foes own body as cover to make them even harder to hit. And the pack wolves should have some sort of teamwork benefit where they are skilled at working together and get a benefit from doing so. And maybe they make trip attacks if they successfully bite a target. And snakes should be hard to see and should inflict venomous wounds. And crocodiles should be adept at moving in water and grapple when they bite a target. Boars should be particularly effective when making a charge attack and perhaps have extra armor when attacked from the front so that they weak to flanking or surprise attacks or something. And bears should maybe have reach with their claws and perhaps the ability to shift between upright and four-legged stances, or something of that sort.
Exactly how you manage that is going to depend on the system. It could be that you can build this by adding some combination of common well-defined features to the stat blocks - features or generic abilities that lots of but not all creatures may have. Perhaps if you are skill based some of the features may relate to skills, though I doubt you could entirely relate the sort of things I describe above purely to a generic skill system or as emergent properties of a skill system without creating a system that was overly complex to resolve.
The more that a bear or a boar or a crocodile is still an end game threat that your characters don't "outgrow" then the more important I would think it is that these non-generic features exist. If they are all just generic stat blocks... yeah, that's suggesting to me some sort of problem with the system.
It's worth noting that my favorite whipping post 1e A&D despite often doing a really good job with giving most things a feature did in fact fall down with regard to this in the general case, as the simple system of AC, HD, and Move just really didn't give you enough to run the creature as robustly as in more modern systems. For example, how stealthy something was really wasn't be a thing, and it wasn't until 3e that you had generic abilities like the 'scent' feature.