It's not about what's "currently in production." It's about what should be a strategic goal to produce the bigger number of tests and use them frequently.
1) The current arguments against the speed tests is that they are not "sensitive" enough. But if one's goal is to do day to day testing and isolating from some environment those that can spread, there's no necessity to detect the RNA leftovers detected by the "sensitive" tests. Rather, the goal is to detect those that are the "powerful spreaders" who in fact have way more instances of virus present in their samples at the moment they are tested. Apparently even the "less sensitive" tests are in fact sensitive enough to catch these who produce a lot of virus. Those who produce a lot of virus have many orders of magnitude more virus RNA present in their samples than those who are in the phase of infection where they aren't so dangerously infective.
Once one accepts that the "ultimate sensitivity" is not necessary, much cheaper tests than those currently used are possible.
2) Even if a single test is still e.g. 20 USD, just stating the goal of having very frequent and regular tests for the sole purpose of protecting a big group of people allows for "pooling" the tests, e.g. testing 20 people together using just one test.
Running the numbers for a university of 1000 people:
1000 / 20 = 50 means only 50 tests per day allow for testing for protection of all 1000 people. Once one batch of 20 is positive, which should not happen every day, additional 20 tests allow for recognizing the one who is the actual carrier. Again, as those that are "dangerous" have many orders of magnitude more virus in the samples, pooling of 20 samples together should be still keep the result detectable enough.
3) Moreover, the test should be less complicated as the one used now: the idea is to test the spit, not to swab the deeps inside of the head.
4) Expensive and slow tests are simply not giving us the possibility to do these "detect early and often" strategies. The really sensitive PCR tests are indeed expensive and aren't as fast as the less expensive can be.
How can the cheap test be made? For example, by designing them to be used the same way the diabetics test their blood sugar levels:
https://www.diabetesaustralia.com.au/blood-glucose-monitorin...
Or even the way the pregnancy tests work:
https://www.123rf.com/photo_107386863_positive-pregnancy-tes...
In short, just new test strips.
Moreover, there are already companies that did some development of such kinds of tests.
It could be done, but it depends on the goals being set.
China successfully used pooling to test the population of 11 million:
https://www.bloomberg.com/news/articles/2020-05-29/how-china...