They quite overlap but while finalement is more neutral, enfin can have a positive meaning. I guess that’s close to the unforeseeable/foreseeable description.
In your example, enfin seuls ! means being alone is a clearly a relief, same with il a parlé enfin (I’d rather say: il a enfin parlé) which imply everyone was looking forward to hear him.
On the other hand, finalement seuls ! is a little heavy, and might be used to just state we are alone after a long period and il a finalement parlé just means “he eventually spoke”.
These adverbs are also not interchangeable in:
Vas-tu enfin sortir de la salle de bains ? (impatience, I want you to leave as soon as possible)
Vas-tu finalement sortir de la salle de bain ? (more neutral, do you plan to eventually leave the bathroom?)
The meaning is identical in enfin/finalement il a tous les vices.
Finalement, ça, c’est ce qu’il dit ! is not very idiomatic and the uncertainty meaning is missing.
Finalement, je dois partir maintenant doesn’t mean anyway, I have to leave now, but I’m unsure either about this meaning in enfin, je dois partir maintenant. I guess the context and intonation plays a strong role here.
Note also that a close expression, sometimes still disapproved (in vain), is emerging: au final. The reason is final being an adjective, le final is accused to break the grammar. My kids are constantly using it anyway, along with grave (for “a lot”), trop (for “very”) and similar.
It replaces finalement (or à la fin):
Au final, il a quand même pu partir en vacances où il (le) voulait.