I've recommended the 1600H to several people (some off-grid), but haven't heard anything like the misery you seem to be having with it. But they're all in cold-water New England, not the warm gulf-coast.
With most of the lower-end tankless units flame-out is common, particularly when incoming water temps are (relatively) warm and the flow rates are low. This is primarily due to their relatively high minimum modulation on the burner (in the 1600 series min-mod is ~30KBTU/hr). If you start the hot water flow at the tap with a fairly high flow it should light up every time without much hesitation, and you can back it off a bit while running, but a burner running at 30K is going to make scalding hot water at flows less than ~1gpm when the incoming water is a tepid 65-75F or warmer (which is common enough in a LA summer). When it can't regulate the temp down, it turns itself off for a period, then re-strikes, which is probably the cycling you're experiencing.
If you just crack the tap a little on startup it'll either not have sufficient flow to run the igniter, or take many seconds. Try just blasting the flow hard for a 3-count before throttling it back, and never take it down to a real trickle- always a good-ly flow. See if that gives you a more favorable result.
Intermittent dishwasher and front-loader clothes washer draws can be an issue for many tankless units in any price & burner range. The flows are often not long enough to complete an ignition cycle, or half the flow per burst is the ignition cycle, and the average water temp is lower than expected.
I don't know how much history they have on the flow-powered ignition system, but I wasn't aware that it had an LED indicator on it(?).