I'm not exaggerating when I say that I have visited stations where the unobtainable list is overflowing and there is not a single available mission offered. Nothing. And these are places where I have some rep with the local factions. I don't think hiding it is going to alleviate the problem and the BB can't possibly be working as intended.
So it's not that you can see them that bothers you, it's that they exist at all.
I agree that the RNG BB can be far too limiting. But I don't think it would make any sense for a bulletin board to only feature missions that you can do.
It's restricted by nature and the solution isn't to make it magically know what you need and present those missions to you.
I have proposed an idea before that should supplement the BB. An extra tab on the BB called "Contracts", where each faction can offer contracts based on type (combat, trade, smuggling, lawful bounty hunting, illegal assassinations and kill lists, mining, exploration etc), where the type is based on the faction itself.
You can browse available contracts, which will give you an outline of what they entail, what rank and reputation you need to take them and the average credit awards. You can sign a contract with that faction at that station and you can only have a limited number of contracts (whatever makes sense).
Once you sign a contract, you can go to the contracts page on the BB of that station, drill down to your contract, which then opens up a list of missions that are limited only to the type that fits the contract.
Your contract is permanent unless you cancel it, the contract expires (eg a CZ contract would only remain valid until the war is over) or you fail too many missions.
This would leave us the BB to get adhoc missions to build up reputation of do odd jobs (exactly as it is now) but we could also align ourselves to a faction and get the specific types of missions we really want to do.
This would really let us define our own career, ensure much less frustration at finding missions, give us a true feeling of place within systems and also significantly improve the military progression model (the two navies would offer contracts to let us work on naval rank % and ascension missions).