It is not a characteristic of Li-ion cells, generally, just the ones with protection. LiCo chem cells will form dendrites if the cell voltage drops below 2.5V, which can short the cell, and if placed back on a charger, they can catch fire or in rare cases catch fire and explode. Other chems of Li-ion can tolerate overdischarge below 2.5V, but it still damages the cell, and enough of that abuse will result in a cell that no longer has much capacity or holds charge, or puts out any amps.
As far as I know, no cell, of any chemistry, primary or secondary gives any warning when it reaches the end of its capacity, though you may be referring to the end of a discharge profile where the voltage rapidly drops down to nothing, though it isn't instantaneous.
But you're making it sound as though not having any warning, when an AA Li-ion protection kicks in and cuts voltage, that it could damage the device you're using, and I'm unaware of any device that uses batteries where that is the case. What CPF members rely on is their familiarity with their cells' capacities, and unconsciously keeping track of how many minutes they've used their lights. Though there were a few early flashlight models with low voltage protection (LVP), in the last few years a lot of flashlight models appeared with LVP built into their circuits. The point of LVP, however, is to protect your Li-ion cells, not to protect the flashlight.
A number of flashlight models have appeared that also have a built-in voltage meter. Zebralight's built-in voltage meter is notoriously inaccurate, but it only indicates 1-4 flashes, and it gives some idea of remaining capacity, though in my experience if the cell is removed and reinstalled, it seems to reset back to 4 flashes even though the cell may be nearly depleted. Any light with Andúril 2 firmware has a decent voltage meter once it is calibrated. When activated, it will tell you the (more or less) resting voltage of your Li-ion to the tenths of volts using two sets of numbers of flashes.
I am not sure I understand your concern. Run AA Li-ion into their protection without worry, that's what it is there for, to make certain you don't damage the cells with overdischarge and have an event when recharging. With enough familiarity with your cells' capacity, you should begin to have some idea of your cells' remaining capacity and just instinctively know when to swap in fresh cells.