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UCI's statelessness is surely not a bad idea. Your example did not prove that (it is a bad idea) but just point out a flawed detail on UCI design.

A stateless protocol means a chess GUI must provide enough information each time an engine starts thinking. In your example, it cannot send enough information about the timer since the protocol does not mention it. It is not a big deal since programmers can solve that issue easily by adding some assumes. Of course, it is better one day we can fix those flawed details in the protocol (version 2?).

I have written engines with both protocols (UCI, WB) and now support them all in my own chess GUI. Thus I have my own ideas about the strong points of each. Both are so good and can do so well their jobs. The stateless idea is the central point of UCI, which makes it a bit more suitable for modern computers and programming - that is why recently it becomes very popular.