Yes, I understand the point and agree with the logic (of course). I still don't think the original statement has been dispelled.

A person 'doing boring things' isn't necessarily a bored person. The experience of being bored is entirely subjective. One person staring at paint dry could be driven insane but another could be meditating upon the physical experience of the passage of time.

The saying 'bored people are boring people' isn't supposed to be taken literally. It is as logically fallible as any other quip in English, I think the difference with this one is that it is used in a more directly negative and judgmental way (and often perceived as such by the recipient).