... and don't remove the 3rd finger until you play the thumb on the second octave, etc...
Chopin is saying don't worry about that, just get the movements smooth.
That's a contradiction for almost all hands except for Chopin who had the most supple of fingers. The poster said that he/she can't predictably play arpeggios fast on white keys beyond the 2nd octave. That leads me to indicate a consistency issue in technique because the stretch is more wider on all white keys, it's more difficult to play an arpeggio more smoothly.
If you don't worry about holding the 3rd finger at all, or remove the 3rd finger too soon, not only you'll get a hiccup, but you'll miss the note because your hand is already in motion, and you might slide pass the note and hit a crack in a multi-octave passage. There is no way to play an arpeggio fast, smoothly, and consistently, both dynamically and rhythmically, if one doesn't try to at least "think legato" in execution. And if you hop from the 3rd to the 1st finger, one is unintentionally liable to play an triplet of ascending chord triads; Even worse, to get a quartet of 16th notes rhythmically now becomes more difficult.
Surely one's not going to hold the 3rd finger in position as long as the 1st or 2nd finger, but just enough to have the thumb either on or very near the position of the right note; Treat it like a pivot without rotation to stabilize the hand in position to transition the 3rd finger to the thumb. This all happens very fast and what I am saying is in super slow-motion.
Every millimeter in motion will make a difference, and I believe the secret lies in the pulling the thumb under the fingers as much as possible toward the intended note without disrupting the transition between the notes both dynamically and rhythmically.