Your baby teeth and adult teeth all began developing before you were even born. Our DNA still contains all the genes that sharks use to grow their endless conveyor belt of replacement teeth, but in humans these genes are deactivated by the 20th week of foetal development.
The advantages of keeping the same teeth through adulthood is that they can be securely anchored in the jawbone, which allows us to chew tough plants and grains.
You said exactly why in your post: "...our biological design..."
There's no such thing. We evolved. That means we're a mix of traits passed along over time by individuals that managed to live long enough to breed.
That's it. That's the whole explanation for any question about "why don't humans do x thing as part of our biology?"
Any given trait is all about lasting long enough to make babies. Once that occurs, all that's left is a general proclivity to ensuring the babies survive long enough to do the same. Regrowing teeth isn't part of that. It's a niche trait that isn't as useful as you'd think for humans. We don't need to gnaw at things, we don't need to crack bones with our mouths, nothing that would make a third set of teeth an advantage, or different teeth an advantage.
Teeth are not easily breakable. We actually can crack bone with our jaws and the teeth will usually survive if the bone isn't too thick; we just have better tools for that because way back when, the proto-humans that used tools had more babies that survived to make more babies. You have to abuse and/or neglect your teeth to break them for the vast majority. There are congenital issues where that isn't the case, but we've also bred ourselves into a social species that takes care of each other, so we aren't limited to a harsh, primitive survival level of things.
I really don't get why people think of teeth as fragile. They're incredibly durable for what we need them for, and require only minimal care to last well beyond breeding age. Even if you factor in modern diets being bad for teeth, regular care for them (brushing and flossing) can stave off those effects for decades. Go search up some of the dental research on old human bodies from archaeological sites. People survived very well with just one set of adult teeth.
And, some humans do have extras that can come in later in life, though it's very rare and comes with drawbacks (according to the last lady I dated that was an anthropologist anyway). Supposedly, having the extras actually weakens the regular adult teeth and makes them more prone to damage. There's always a tradeoff in things like this.
Just a note, biology doesn't have a design. If you're looking for some kind of logic or plan, you'll be disappointed.
Things are the way they are because a long time ago, it helped something survive and procreate. That's it, survive and procreate. Every other consideration is secondary.
We can theorize about why two sets of teeth were advantageous at some point, but that doesn't provide an answer to "why?"
The diet that we evolved to consume (fruits, lean meats and fibrous plants) was much less damaging to our teeth than the current high-sugar, high-fat, highly processed foods. And human lifespans was shorter, so less time for teeth to damage.
So there wasn’t a strong evolutionary need to regenerate them (unlike an animal like sharks)
I mean.. we grow teeth a total of 3 times. The first for our baby teeth, the second time for our 'mature' teeth, and the make up 'wisdom' teeth to fill any that might've fallen out at that point. I'm guessing those three growths were the most needed for humans early survival before we got all fancy with farming and hygiene. At which point we kind of broke survival of the fittest and things just kind of happen now.
Kind of like how humans are one of a handful of mammals that didn't evolve out of menstruation.
I think they are intended to, and they actually do... once (child teeth). Probably just broken due to genetic decay or environment (e.g. if humans are no longer fully maturing and what we call adult teeth are actually "intermediate" teeth). I suspect a deeper understanding of the recent tooth-regrowth drug(s) may provide a clue as to why it is currently broken.
My personal theory is that this mutation among humans would lead to older members of the tribe living longer and being more of a burden on the younger members.
I think there just hasn't been an evolutionary need for them to regrow. In past millenia, people had kids and died before toothlessness really became an issue and teeth lasted longer before our modern industrial diet.
You're saying we weren't intelligently designed? Heathen! Someone get the "burning people" sticks! Quick, before they reproduce amd spread their frail-teeth genetics!