Yes, it is not desirable to run the battery down, whether "battery saving" is on or off. Battery saving merely limits the full charge, so that the battery is only charged to 80% of full charge (or 90%, whatever the limit is). The general guideline for lithium batteries is to not fully charge, or discharge. These are only guidelines for long term usage, as they will not turn into a pumpkin if you break the guideline. Also, they have a finite life, and there will be manufacturing defects. They will degrade and die even if you follow all the guidelines, or do not even use the battery. So there is not much use in over worrying, or thinking that you can keep it running forever.
It is just a good habit to follow best practices to lessen the stress on the battery, and to try to lengthen the battery life. The things to try to avoid are situations that cause high current/heat. A deeply discharged battery charges at high current, which generates more heat. Bad heat dissipation, like having it in a thick case, sitting on an insulated surface (like paper, books) allows the heat to build up. Taking it out of a case is not practical, but it is easy to put the device flat on a metal, or other hard surface that will dissipate the heat better. Or at least not put it on an insulating surface, like magazines, books, foam, etc. Propping it up on edge (leaning against something) while charging so that the heat dissipates into the air better is another way. There are many simple things that can be done.
Having a small fan blowing on it would help. Personally, I do not do that, but I do put a cooling fan on my Router, in the belief that cooling is always better for any device.
It is like the recommendation for people to not smoke, eat healthy (hard to determine what this means) , and get regular exercise. Some people smoke all their life, and are OK. Some people never smoke, and run into problems. But following the best practices, is still something to try for.