As a few threshold considerations:
How do you determine what “information” is relevant? And at what level of detail? I could zoom in on any number of areas of inquiry about John’s original “instantaneous” journey and come up with additional information until the cows come home. In doing so, it seems that the “time” conclusion would change based in what information I deem relevant.
As another big issue, I would ask what qualifies information as being “available”. In other words, available to who?
It seems to me that your proposal gives ad hoc answers to these questions which fit your desired outcome.
More generally speaking, I think you need to fold a whole lot more subjectivity into a conception of time. In my view, time itself is a quale.
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