I'd say that this is partly the case. Because of the rounds, even a single block operation will be operated on using multiple times. A block also doesn't take any special input: you have one block that is mapped to one output block.
However, that's not the complete story. First of all, some of the calculations in the rounds may require a bit more magic than usual. If there are any possible edge cases then those may not be triggered by one single block operation. Those you should test separately.
More importantly: while the above tests the block operation, it doesn't test the streaming operation. This is much more likely to cause issues. The problem is that encoding / decoding (e.g. bytes to words) and buffering are less trivial than that many seem to expect. So it makes sense to test all kinds of plaintext sizes, for instance, and make sure that update
functions perform correctly. This is not so different from testing non-cryptographic functions; you'd expect the same issues with encoding/decoding operations and I/O for instance.
One issue that I've seen many times is that testing teams ignore the implementation itself. It is all fine to test against official test specifications, but if there are implementation specific design choices made, then these should be tested as well. As an example: it could well be that an implementation has a buffer overflow even if all official tests run without incident. I'll not repeat any horror stories here, but yeah...
Finally, as this is a cryptography site: if you make any claims about protection against side channel attacks then you should probably test those as well. Of course you're somewhat lucky here as Bernstein generally assumes these kind of attacks when designing a cipher - but implementations may be vulnerable none-the-less.
All this said, it never hurts to run your implementation against a more extensive set. Just take any existing library (or two independent ones) and generate a large set of vectors yourself. Again, focus on edge cases; doing a bit of fuzzy testing never hurts either.