Is saying that people dying from disease isn't as bad as people dying in a mass shooting an "Appeal to Nature?"

I don't think so. The appeal to nature is unreasonable because it is the belief that natural is better than unnatural because of its "naturalness." I think "only natural," in this context, is shorthand for "we have no/little control over," or even "it is and always has been part of our lives." Therefore, the argument is really that we shouldn't die because of things about which we can relatively easily do something. However, if the argument is actually that it is better to die of something "natural" like childhood cancer rather than something "unnatural" like getting hit by a bus at age 105, simply because childhood cancer is "natural," then it would certainly fit under this fallacy.

I also think you need to rethink your position... people certainly made a "big stink" out of people dying of natural COVID disease yet the million+ people who die in unnatural car crashes each year go unnoticed by the average person. People tend to get concerned about how people die when that number increases significantly and rapidly, especially when the numbers vary greatly among societies. This typically indicates some kind of public policy that can be changed in order to prevent deaths (like mass shootings).