is it safe to rely on the built-in safety PCM of a Li-ion battery?
A better question might be, "is it safe to rely on the PCM in a UN 38.3 approved lithium ion cell or a battery constructed of such cells?".
And that answer is definitely yes. UN 38.3 approval requires a fairly extensive set of safety tests that a battery cell must pass. Furthermore, even if a battery is constructed from UN 38.3 approved cells, the battery as a whole still must be retested to receive its own UN 38.3 approval.
Among these tests are overcharge and overdischarge tests. They involve charging or discharging a battery/cell with a charger that, with no protections within the battery or cell, will charge or discharge the cell at twice the manufacturers maximum charge or discharge rate for 24 hours. The cell is then monitored for 7 days after this abuse to check and see if it does anything dangerous, like catch fire or explode.
In the case of Lithium Ion cells, the only way to ensure that this test doesn't result in a fairly spectacular test failure is a working PCM that reliably cuts off the cell from external loads or power to prevent damage.
UN 38.3's primary purpose is to ensure that batteries can be safely transported on air or ground freight. For example, in the US, it is illegal to ship lithium ion batteries that don't meet UN 38.3 approval except by ground, and even then, this must be declared and usually an additional hazard fee is charged by the carrier.
Simply put, you shouldn't ever use a lithium ion battery that doesn't meet UN 38.3 unless you have no other choice and are very aware of the risks and additional regulations that will apply.
If a cell or battery meets UN 38.3, then you can expect to rely on the PCM. If it is good enough for the United Nations Transportation Department, it is probably good enough for any commercial usage case. That's kind of the point.
The battery you linked is UN 38.3 approved, so that is a good choice and you can expect the PCM to do its job.
does it damage the battery to actually reach the built-in safety PCM and can it be then safely charged again?
Think about this for a second. If the battery falls below the cut-off voltage, the PCM allows the gates of two back-to-back (common drain) configured MOSFETs to discharge. Any stray charge has a conduction path through a comparator-actuated transistor inside the PCM IC, meaning that those MOSFETs are not coming on again unless the battery voltage exceeds the cutoff voltage plus hysteresis.
But such an increase will never occur, because there is no longer a working current path into or out of the battery. You can't discharge it. You can't charge it. There is no way to get any energy into the cells to raise their voltage. At least not without disassembling the pack and cells and removing the PCM. And you are not liable for the results of such activities. If someone really wants to circumvent your protection, they can, but if they do so, it is also their responsibility, not yours.
So let's be clear: if the battery gets discharged low enough to trigger the PCM, either it was a transient drop due to a heavy load, and the terminal voltage recovers enough to be higher than the cutoff and the hysteresis, then it can be charged again. If it recovers to a higher terminal voltage, it was not really discharged below the cutoff, but rather ESR voltage drop made it seem like it was. Hysteresis ensures that only when this is really the case, can the cell return from cutoff.
However, if the cell is really over discharged, or like in your example, the user leaves it uncharged for a year and it self-discharges below the cutoff, it's gone. The PCM will have cut off all current paths to the cell, so there is no way any charger can even charge the cells to raise the voltage again. The voltage cutoff is set at a point that the manufacturer of the cell has chosen as the lowest safe voltage to charge from. If a cell discharges below this voltage, it means the manufacturer considers it unsafe to charge it again and the cell is forfeit. The PCM ensures it cannot be charged again, and this is the intended outcome. The user is forced to replace the battery, which is the correct action to take.
Whether or not it really damages the battery or not is not really important. What matters is the manufacturer believes it could damage their cell, and it isn't worth taking chances. And the answer is no. It cannot be safely charged again. Best practice is to simply consider an overdischarged Lithium Ion cell as destroyed.
If this sounds weird, it isn't. I have personally experienced the self-discharge death of a laptop battery pack because I forgot to leave any charge in it, let it sit unused for a year, and low and behold, the terminals had gone high-z (mosfets turned off). It couldn't be charged. It couldn't be discharged. It was gone.
This is a functionality that exists in most protected lithium ion cells, and letting a device sit unused without charge for a year or two will destroy the batteries in most devices.
would it be good design to do everything possible (using a voltage supervisor, for example) not to reach the built-in safety PCM of the battery?
Yes. Because the battery will effectively be destroyed from the viewpoint of the user, and from the safety stand point of the company that made the cell the moment it falls below the PCM's cutoff voltage. The PCM cutoff is only ever going to happen once. There are no second chances. It is the final and ultimate safety measure and when it engages, it is permanent.
You don't need to avoid relying on it for safety reasons though, it is quite safe and will work. You don't want to rely on it because you only get to rely on it one time. And after that, the battery is toast. Safely toast, but toast all the same. So I guess it depends on your priorities.
does a better strategy exist?
Not really. At the end of the day, you're at the mercy of the user and the battery's own self-discharge, and there isn't really anything you can do about it. Lithium Ion batteries cannot really be expected to survive more extended periods of time in a mostly-discharged state.
Your only options are really more or less what you're already doing: try to warn or otherwise prod the user into preventing the untimely death of their battery when it starts getting really low. Then, you can increase the chances of survival a bit by having a second voltage cutoff where the gas gauge and whatever else you have attached go into deep sleep mode and stop working/warning the user. This reduces the drain to just the self-discharge and whatever the drain of the PCM is (but, typically, this will be very small in comparison to the battery's self-discharge).
After that, you've really done all you can. It is up to the user now, and beyond maximizing the time they have to start charging the cell again by reducing the discharge to the minimum (which is, for all intents and purposes, the self-discharge rate of the cell itself), it is out of your hands.
It isn't ideal, but it also mostly works. It is far better that one's forgetfulness destroys $100 of lithium ion batteries than that same forgetfulness (combined with allowing overdischarged cells to be recharged) instead causing a fire and many thousands of dollars of damage, or worse. And I say this as someone who actually managed to destroy a pretty expensive laptop battery. I'm glad it didn't let me charge it again.