I think it's sad for the reasons you mention, and the people in the comments section were probably very moved by all that. For me, the video was like a tragic reminder that time renders us all to nothing.
Why is it sad to watch this video? Is it sad that two conscious beings are attached to each other, and soon the attachment will end? Is it sad because we are reminded of our own impending doom? What fundamentally invokes the emotion we see in the comments section of this video?
I like that the 2 reasons you mentioned are so different. The first involves empathy, and the second is selfish. I'd say the reason is essentially the second one for everyone although most may not be consciously aware that it is. I think service to others (STO) is just another way for service to self (STS). STO things like empathy, or loving anyone else besides yourself, even in their purest forms are illusions and are 100% selfish; they're based on how the other person/thing makes you feel or provides for you. It's easiest to see in the behavior of babies and children, when they are closest to their original selves or 'id'. What benefits you or your offspring is all that matters. Doesn't necessarily make it any less beautiful though.