Cancellation of these Voyages is not as simple as it seems. Since there are people who do like these, myself included, you can't really just make it so one player can elect to cancel them off - that could have an impact on someone intending to try and do them when you start talking Crews. That said, there are some options here I would think:
Option 1: Once one (or more) Bottle Voyages are picked up a new item is added to the Voyage Table on the Ship such as a Stack of Papers with the name of "Supplementary Voyages" that can be viewed by the Crew (or other Pirates) to see what is there, and for the Crew having an option to "Vote to Cancel" just like a regular Voyage. This presents the option to clear them without making it so one member of the Crew can dictate if these Voyages will be done or not while letting other Crews get a further sense of what you might be getting up to in addition to what they can already see on your Voyage Table (and Map Table). This options would require new world assets and new interface options to be setup but they are at least related to existing systems to reduce implementation complexity.
Option 2: Make the Bottle Voyages be individual to Pirates rather than belonging to the Crew. Then, as an individual, I see no reason why you shouldn't be able to have control over your own list of Bottle Voyages as you see fit. An added bonus of this is it adds another reason to engage with the "Show Map" mechanic that we have within your own Crew as to get the Crews help with these you would need to actually show them your Map/Voyage details. Thus it encourages player interaction. This method would take a little bit of work in changes to the distribution mechanics as well as implemention of various management options, but again they are based on existing systems.
Option 3: Keep these as Crew Shared Voyages and keep them unable to be cancelled in a session, but give them their own Radial as a sub of the Quests Radial (much like was done with Notes). In this way, they don't clutter the main Radial which seems to be the biggest point of contention as it pertains to these, but otherwise leaves the mechanics behind them alone. Being just a UI change in effect should probably make it easier to accommodate, particularly considering the underlying framework for this already exists for other functions.
I suppose we could say there is a fourth option here in doing nothing at all and just leaving the system as is since it is, at most, a minor personal inconvenience that is reset every new session. This is the worst response to it, since I do agree this seems to be an area where improvements could indeed be made that would enhance the overall player experience through what seems to be little, albeit not non-existent, efforts. Honestly, any of the above options presented would be an improvement in my eyes, and I could be comfortable with any of them if we made some changes.