Here's what I interpret...
Mom wanted to vent to OP in hopes he would sympathize with her and talk to his sister. This would be in hopes that she would change her ways and think about what her mother is going through, and, offer to help (for a change?). Obviously, things didn't go that way.
OP instead sounded like a detached entity who decided to tell his mother that she needed to properly communicate with her daughter instead of venting to her son (who we didn't know initially didn't live anywhere near home for a very very long time, which is also irrelevant to the conversation since mom didn't ask for his help). This comes off as unsupportive in general and crass when he brings up the Karen comment. I can say without a doubt that what you said to your mother would push her away. If you want that, keep saying things like you have, and also never apologize for what you said.
It's OK to mention that you think it may be a good idea for your mother to tell your sister exactly what she told you in hopes her daughter would understand. That's constructive criticism.
For example, if you said almost exactly what you said, but worded it: "Mom, I understand where you're coming from. Clearly she isn't taking into consideration what you're going through. Maybe it would be helpful for you to tell her exactly what you told me." You could also offer to talk to your sister as well, and include, however, that it may be more effective coming directly from her.