Ive got a one wire in mine, and about 50% of the time, it doesn't start charging right on startup. As with some of my other 1 wire converted machines, it only takes a little blip of the throttle to get it going, maybe 1k engine rpm for a second or two, well within the capability of even a cold engine, though I generally don't even bother to blip it until after my walkaround.
Many of those references (ive seen them too) to the 2k rpm are, I think, referring to the alternator shaft speed, not crankshaft speed/ engine rpm. Bearing in mind that these alternators like virtually all others, are overdriven from crank speed, it shouldn't take much to get it past 2k.
If you have to wing the engine past 2k to get it charging, I think you're facing a dying alternator, and the flashing switch is, while quite useful, just prolonging it a bit.