Well, I first apologize for not understanding the post as I had never heard of the word 'disinterested' and as soon as I read it here, I thought it was a word that the person asking the question had confused with "uninterested" and therefore, wrote something that came to my mind about using the prefixes wrongly (dispossible). Now, I just searched and found out that the word 'disinterested' does exist and it fits the best here.
It does make sense to use the word 'disinterested' for someone being concerned about others' well-being, as the word itself means to be unconcerned about the outcome of something.
The person asking the question is confusing 'disinterested' with 'uninterested' and therefore, thinking that it is impossible for one to be showing no interest in something while being concerned about it at the same time.
Someone 'uninterested' is a person who doesn't take interest in a particular thing while someone 'disinterested' is a person who doesn't care about the consequence of something and acts neutrally, without showing any concerns about getting benefit out of something or experiencing any loss out of something.