The exact dates quoted in popular science articles are typically not what you find in scientific papers. There you have a lot of data points and discussions of confidence intervals, etc. To address your pottery example: Indeed, if we had dated only one shard, we could not conclude much from it. If instead we had 10,000 shards over the whole area of a city and find that they suddenly stop at a certain date, we can be fairly sure that this date is very close to the destruction of this city.
The general problem with biblical archaeology is that it is particularly susceptible to dogmatically motivated biases. During my studies in Heidelberg in the 1990s, I also took a few courses in this field. I never met an archaeologist at the entire university who even remotely believed in the existence of king David, let alone Solomon, Saul or even Joshua or Moses. However, when I visited Israel with a group of students, we were proudly presented with artifacts from these "empires" in some kibbutz museums. We or our professors did not directly object, of course, but smiled politely, just like in the churches where we were allowed to admire the miraculous icons.
The general problem with biblical archaeology is that it is particularly susceptible to dogmatically motivated biases. During my studies in Heidelberg in the 1990s, I also took a few courses in this field. I never met an archaeologist at the entire university who even remotely believed in the existence of king David, let alone Solomon, Saul or even Joshua or Moses. However, when I visited Israel with a group of students, we were proudly presented with artifacts from these "empires" in some kibbutz museums. We or our professors did not directly object, of course, but smiled politely, just like in the churches where we were allowed to admire the miraculous icons.