Usually on the stuff I dye dark, I'll apply Minwax "special dark" paste wax and work it into the pores to eliminate the "white". I'd suspect your Odies is doing the same.
Fun fact about your "tea"
You can also use a standard Lipton tea bag (a.k.a Liptorn). Just dip it in a cup with enough hot water to saturate the bag and start applying like you would with a towel dipped into stain.
Helps if you have a box of the buggers from the 80's stashed in the back of a cabinet. 😝
If you want a black with some depth into the wood, the tannin booster is a great technique!