We actually have a lot of archaeological evidence from Carthage-- the bigger issue is that we have little to no written evidence. Let's go through the history of Carthage and catalog what we have:

First, from the earliest days of Carthage, we have a little bit based mainly around Byrsa, the main hill where Carthage was allegedly founded. From here, we can partially reconstruct what the city looked like: some walls, a cemetery, maybe some houses. However, this first layer was buried when Carthage was renovated and expanded as they rose to power: the cemetery was emptied and workshops were created, the walls were expanded, and new buildings were created. This is the point we usually consider the "classic" Carthage that is the most interesting thing to study.

Unfortunately, Carthage lost the Punic wars, which led to drastic changes for the city. First of all, the city was mostly destroyed when it was captured by Rome at the end of the third Punic War. While they didn't salt the ground afterwards, the city was mostly abandoned for the next hundred or so years. Eventually, the Romans decided to settle it, as it was near excellent farmland and a great port, but they made really drastic changes.

The first major change was that the Romans demolished Byrsa. The hill. Made of earth. That's a big deal. When the Romans completely destroyed the center of ancient Carthage, they utterly rewrote the landscape of the city. The second major change was them literally rewriting the landscape of the city to align along a Roman colonial plan. Interestingly, rather than going north-south/east-west, like they preferred, the Romans aligned the cardo and decumanus alongside the coastline. The Romans were rewriting the streetplan as if Carthage was just another piece of land that they had captured, without previous history, and buried the archaeological evidence even deeper.

Now to get to the issues with retrieving this evidence. In a lot of cases, the archaeological evidence may not be lost, but the desire to unearth it isn't there compared to Greece and Rome. First, Greek and Roman art has been coveted for millenia, and still much of it has been lost. Similarly to the Etruscans, very little from that era has been saved without a centuries long intent. Along these lines, there hasn't been the funding to support archaeological studies on the same scale as Rome-- mostly because that would be absurd. The final major problem with Carthaginian evidence is the lack of written evidence. Barely any of Carthage's literature survived, while the writings of the Greeks and Romans are perhaps their greatest legacy. A lot of their culture was lost from this.

But now onto the good news for archaeology in Carthage. A lot is still there, from both pre-Roman and Roman Carthage. The old harbor is still visible, which is the most identifiable and one of the coolest things Carthage ever made. We have dug down through the layers of Byrsa, which is how we know so much about it, and found a lot of information about the history of Carthage. This isn't to mention the tophet and all of the archaeological evidence surrounding it. We've also found a lot of religious stele, which we can date through various periods to understand how the religion changed over time. We also have been able to find sites outside of Carthage itself. One recent (and ongoing!) archaeological project has discovered several sunken ships from the first Punic War near the Aegates islands. This was the final battle of the first Punic war, where Carthage attempted to launch a last disk effort and failed. So far, we've seen a lot of weapons, interestingly mostly Roman, and we've learned and hypothesized a lot about classical warfare from the findings.

So it's not like Carthage has been lost to history, just less prevalent than Rome or Greece. The two biggest reasons are the lack of literature and because Rome built over Carthage, but this doesn't mean we don't have any information. We're discovering more and more every day.