I admit that it still feels a little counterintuitive to me
Hang on to that feeling, it's your guide to the truth.
In the situation you describe, there is no stream of consciousness which starts where you are now, and which then jumps discontinuously to the stream of consciousness subsequently occurring in the copy's body.
At most, there is a new stream of consciousness, initiated in the newly created copy, which begins with the (illusory) feeling of a jump, because the first moment's experience is being matched against implanted short-term memories.
To believe otherwise is to believe that a single connected stream of subjectivity can supervene on physically discontinuous systems.
Unfortunately, the way people are likely to resolve this problem is to totally deconstruct subjective timeflow and say that in reality, there is no stream of consciousness anywhere ever, there are just disconnected self-moments in the timeless multiverse, et cetera. But in that case all forms of anticipation are illusory.
(My position is that time and change are real, and there is such a thing as continuity of existence in time. Many people here have a problem with that, because of many worlds, and because even single-world physics describes space-time in a static, geometric way. But that's just a limitation of our current conceptual tools, and not a refutation of the phenomenon of time.)
I conclude, after reading this a few times, that I don't really know what you are labeling with the phrases "stream of consciousness," "stream of subjectivity," and "subjective timeflow." I don't know whether you mean to refer to one, two, or three different things, nor how I would recognize that thing (or those things) if I found an instance of it (them) in my oatmeal, or how I could tell if I subsequently lost it (or them).
That said, if it's what I ordinarily understand people to mean by "stream of consciousness", ...
The subject of copying people and its effect on personal identity and probability anticipation has been raised and, I think, addressed adequately on Less Wrong.
Still, I'd like to bring up some more thought experiments.
Recently I had a dispute on an IRC channel. I argued that if some hypothetical machine made an exact copy of me, then I would anticipate a 50% probability of jumping into the new body. (I admit that it still feels a little counterintuitive to me, even though this is what I would rationally expect.) After all, they said, the mere fact the copy was created doesn't affect the original.
However, from an outside perspective, Maia1 would see Maia2 being created in front of her eyes, and Maia2 would see the same scene up to the moment of forking, at which point the field of view in front of her eyes would abruptly change to reflect the new location.
Here, it is obvious from both an inside and outside perspective which version has continuity of experience, and thus from a legal standpoint, I think, it would make sense to regard Maia1 as having the same legal identity as the original, and recognize the need to create new documents and records for Maia2 -- even if there is no physical difference.
Suppose, however, that the information was erased. For example, suppose a robot sedated and copied the original me, then dragged Maia1 and Maia2 to randomly chosen rooms, and erased its own memory. At this point, neither either of me, nor anyone else would be able to distinguish between the two. What would you do here from a legal standpoint? (I suppose if it actually came to this, the two of me would agree to arbitrarily designate one as the original by tossing an ordinary coin...)
And one more moment. What is this probability of subjective body-jump actually a probability of? We could set up various Sleeping Beauty-like thought experiments here. Supposing for the sake of argument that I'll live at most a natural human lifespan no matter which year I find myself in, imagine that I make a backup of my current state and ask a machine to restore a copy of me every 200 years. Does this imply that the moment the backup is made -- before I even issue the order, and from an outside perspective, way before any of this copying happens -- I should anticipate subjectively jumping into any given time in the future, and the probability of finding myself as any of them, including the original, tends towards zero the longer the copying machine survives?