For safety during transportation and storage, from the factory to the warehouse to the store, one would like the battery to be mostly discharged so that it's holding less stored energy in case of some mishap. If a forklift spears a big box of product, you'd like a minimal amount of energy released from the dead short which may be caused by crushing or puncturing cells. Probably there will be some protection built into both the cells and the product to try to prevent fire, however it's still safer to have less energy to release. Airlines in particular may have rules or laws which describe how batteries should be transported.
For purposes of maximizing lithium battery lifetime, it's better to keep them charged between 20-80% of full capacity most of the time. It's very damaging to over-discharge a battery and may render the cells useless. While any product shipping with a lithium battery should have circuitry to prevent over-discharge, it's possible the load on the battery will never quite be reduced to zero even when the product is "off" or in full shutdown, as there could still be a low-power microcontroller watching for a press of the power-on button, some tiny current powering the battery protection circuitry, etc. While this circuitry might run "indefinitely" from a full battery, if the battery is run down to empty and then the product is left in a drawer for a year it's possible the battery could get over-discharged regardless of the circuitry trying to prevent it.
One possible scenario could be that the factory charges the battery, the device goes into inventory for over a year (due to the product not selling well), the battery loses most of its charge by self-discharge and a small amount of current powering the "off"-mode circuitry e.g. microcontroller. Then if the device is turned on with this nearly-dead battery in it, the large pulse of discharge current causes the battery voltage to dip to where the undervoltage protection triggers, which may or may not turn off the current before the battery has been run below its minimum voltage for some small amount of time, like milliseconds. It probably won't destroy the cell but isn't a good idea. Especially if the user sees the product doesn't turn on and keeps hitting the power button a dozen times before deciding it must need to be charged up.
To minimize tech support phone calls, it's better to tell the user to fully charge their battery before trying to use the product, so you don't get calls about the product "not working" when the actual problem is that the battery is simply empty. It's one less variable in debugging whether the product is faulty. A product may have features that take more power than the baseline mode, and might turn on and operate properly, but then appear to fail when a higher-power feature is turned on (when in fact it's shutting down or resetting). Turning on the backlight in an otherwise simple device, or wireless transmission, loud audio, fan or motor, anything that's a large percentage of power consumption. If the battery is fully charged, it will have enough energy to power all modes, so this problem is avoided while talking to tech support.
That said, there's a lot of mythology out there regarding how batteries should be handled. The customer care person is probably not an engineer, and may be going by some mix of training, passed-around heuristic, and made-up baloney that's designed to mitigate unhappy boss or unpleasant customer rather than detect "the problem".
Best Answer
You asked:
The curves you supplied cut off at 3.0V - so suggesting 2.8V is below what the sourve of that graph thought was wise.
3.0V is a sensible lower limit BUT as you graph shows, the end point voltage depends critically on load. The higher the voltage the longer the cells will last, all else being equal. If you discharge to 90% to 95% of available capacity at a given load you lose little energy and improve battery lifetime. Importantly in some cases, having a reserve means that if there are stray loads on the battery it is less likely to be discharged to a dangerously low point that it can not recover from.
You didn't ask, but at the top end you can also make capacity-lifetime tradeoffs. The chart below is from battery university. This information is not usually presented and can be immensely useful. The "capacity at cutoff voltage is the capacity achieved when you swap from constant current to constant voltage charging at the point shown.
Again, using less than full capacity extends battery cycle life.