It's an online algorithm. It's meant to be used on essentially infinite streams of data, such as a live dashboard of latencies of a running server. In that context, providing a 5 element, or any finite list as an example seems like a non sequitur.
Your examples do relate to problems the algorithm actually has in this application, but they manifest as things like "extremely large warmup time" and "adjusting to a regime change in the distribution taking time proportional to the change". For instance if your data is [1000, 1001, 1000, 1001...] then it takes 1000 steps to converge, which may be longer than the user has patience for.
However, the algorithm does always converge eventually, as long as the stream is not something pathological like an infinite sequence of consecutive numbers (for which the median is undefined anyhow).
Your examples do relate to problems the algorithm actually has in this application, but they manifest as things like "extremely large warmup time" and "adjusting to a regime change in the distribution taking time proportional to the change". For instance if your data is [1000, 1001, 1000, 1001...] then it takes 1000 steps to converge, which may be longer than the user has patience for.
However, the algorithm does always converge eventually, as long as the stream is not something pathological like an infinite sequence of consecutive numbers (for which the median is undefined anyhow).