If we drop the common-support-$K$ condition, and instead of requiring uniform convergence of derivatives on that common support, we simply require uniform convergence on all of $\mathbb R^n$, ... which might seem reasonable, and simpler, ... the space of test functions is no longer (sequentially) complete, which would be undesirable.
This incompleteness is similar to a simpler example, that of continuous, compactly supported functions with a single norm, the sup-norm over the whole $\mathbb R^n$. This space is not complete with respect to the corresponding metric: it is a standard exercise that the completion is the space of continuous functions going to $0$ at infinity.
Similarly, using sup-norms of all derivatives over the whole $\mathbb R^n$, the completion of test functions (with the corresponding metric attached to this countable collection of norms) can be shown to be the space of smooth functions so that they and all derivatives go to $0$ at infinity.
The "correct" topology on test functions (or even on continuous, compactly-supported functions), "correct" in the sense of being suitably complete, is more complicated than Hilbert, Banach, or Frechet, called "LF", for "(co)limit of Frechet".