The obvious answer is that the Romans deliberately wiped out all traces of them. But can that really account for the shocking lack of artifacts from a major world empire? For example, there should have been Carthaginian goods traded all around the world.
There's actually a lot of Carthaginian artifacts. They just don't get much attention because they look very similar to Greek artifacts and other artifacts around the Mediterranean.
Don't forget that the Carthaginians were Phoenicians, and so much of their early material culture is very similar to other Phoenician material excavated all around the Mediterranean, from Tyre all the way to Spain (the Phoenicians founded Cadiz as well, according to Strabo). Excavations in these colonies have given us many artifacts such as this statue of the Phoenician god Melqart, who is conflated with Hercules in later Phoenician religion.
So many Phoenician artifacts from Carthage look like either artifacts from Phoenicia or like artifacts from the region where the Phoenicians colonized. The same goes with Carthage.
And by the same token, a lot of artifacts exported from Carthage to other places would often be understood to be "Phoenician" artifacts, and there wouldn't always be a direct connection to Carthage specifically. So we can't say that there's really a lack of Carthaginian artifacts, just that they're not always called Carthaginian.
In fact there have been numerous excavations at Carthage, including the famous Tophet, the burial ground that appears to be evidence of human sacrifice (I wrote a bit about the evidence for that here in this thread). Some of the preliminary excavation reports from the more recent excavations are actually online and you can read them if you wish:
Another article outlining the results of many years of excavation can be read here on Academia.edu. This author makes a note that the residential buildings at Carthage were found to be very similar in style to those typical of the Southern Levant (the ancestral homeland of the Phoenicians).
In fact, much of their art is reminiscent of Greek and Roman art, the two cultures that dominated the Mediterranean world for much of Carthage's existence. Since Carthage itself was Phoenician, they too were quite influenced by Greek style. You can see a number of Carthaginian artifacts at this website. On this forum there is a dump of images of Carthaginian and Phoenician art (I haven't read the posts, just using it as an image dump, I do not necessarily endorse the opinions of the people who posted there). It looks pretty generically Greekish/Romanish, doesn't it?
Much of their early art wasn't really of people though, but displayed religious iconography like this statue showing symbols of Tanit, the main Phoenician goddess of fertility and the sea, or this other one from the tophet (the child sacrifice graveyard) of Carthage itself.
TL:DR
There isn't really a dearth of Carthaginian artifacts, just they aren't as sexy as the Greek and Roman ones of similar time periods, so they don't get as much air time. In addition, they are usually called Phoenician artifacts in the earlier periods of Carthage's history, as opposed to Carthaginian artifacts, especially when they are exports to other places.
