They could easily be writing in to me to say that it’s up to your mom, not her kids, to be the answer to her loneliness, but one sibling is pressuring them to be your mom’s entire social life. Obviously, we all need to spend time with people we love, for their benefit as well as our own — and I’m not suggesting that everyone let Ma figure it out her own darn self.
But, it’s OK to give your mom some responsibility for her own happiness before you go blaming your sibs. It also would help, no doubt, if you looked at this not with a conviction that they’re wrong and you’re right, but instead that they have their way and you have yours. That makes your task a lot less presumptuous. Plus, if you work with their strengths, then you'll likely be much more successful.
But, again, start with Mom herself. Talk to her about what she needs, what she craves, how her arrangements at home can be adjusted to make these things easier. Ask what her hometown offers in the way of support and socialization. Ask if she wants you to look into it for her. Does she even want you doing all this?
Then, with her or on your own, figure out which sib has a strength or skill — who’s mobile, who’s organized, who’s plugged in, etc. — that can be leveraged to enrich your mom’s life. Then, if Mom won’t do it herself, you approach each sibling in a “Hey, I have an idea” way – or better, “Mom asked me to ask you if you’d be willing to [blank],” which is going to be so much better received than “I'M TAKING CARE OF MOM ALL BY MYSELF, YOU INGRATES.”
As in: “Mom needs help with remembering appointments and coordinating rides. Are you willing to step in as schedule-keeper? It’s doable from home.” Or, “Mom said she wanted help sorting old photos. Do you have a free day sometime in the next month?” Specifics are easier for people to grab onto, especially if they’re expressed in a way that takes your family dynamics into account.
Don’t rule out professional guidance here, either. A geriatric social worker might yield useful ideas, and you can easily start the process of finding one at eldercare.gov.