To clarify the nature of the mystery let me ask another question.
How do the esters and complex hydrocarbons that give one one's sense
of smell get from the mouth to the roof of the sinuses where the nasal
hairs come through the skull from the olfactory bulb into the top of
the sinuses?
I am not sure how the tannins get from the mouth to the sinuses, but
they evidently do.
The effect is topical, not systemic.

By all means use tea if that makes you more comfortable, I have found
very strong tea with a little salt and lemon is considerably better
than any other irrigation solution other than the pecan based ones.
Black tea contains the most tannins, but the tannin in tea pale in
comparison to those available from the humble pecan.
Pick out the nutmeats and grind the shells in a blender or coffee
grinder. Sieve to remove some of the fine red powder
and reserve.
Put the fine ground shells in a coffee filter
with the peal of an orange and a couple of cloves as an option, and
perk with one cup of water per 1/4  cup of ground shell.
Brew as you would for coffee.
You can add salt just until it tastes slightly salty, and lemon juice
at this point, and use it as an irrigation solution, sweeten it an use
it as a diffusion solution or
for something that is a bit more effective don't put anything in it
yet:
Put in it a blender and add nutmeats and the fine red powder back to
the brew and blend until creamy, then blend twice as long.
Add the salt, 1 pk of cherry coolaid (red dye that adheres to the
bugs, and V C) to 4 cups of solution, or 1/4th pk per cup, and add
splenda to taste.
Hold it in your mouth and then lay down inclining your head from one
side to the other, about 10 minutes each side will pull out quite a
bit of junk.  Longer when you are just starting. And when you first
start pulling the junk out, the removal will give probably give you a
headache, so take some advil before you start, and for me the mucus
that I pull down "causes emisis in the primate model".  But it has
with everything, untreated PND, antibotic washes, and to some degree
with the tannin just a bit less so. The tannin will cause the mucus to
shrink, but plan on spitting out solution, mucus and all out every 10
minutes or so.

to get an idea of the relative properties of tea, coffee and tannin
prepare a cup of each and let sit on an undistrubed spot for
a couple of weeks. In my experiment, The coffee showed mold. The tea
showed mold. The pecan solution has not shown any growth of any kind
so far after 4 weeks. Pecans have been defending their babies against
the micro-gang forever. They got it right.
But do your own experiment.