If you are not the executor you do not need a solicitor to sort out probate. Even the executor can sort out the paperwork themselves if they prefer, using a solicitor isn't compulsory and you/sis definitely don't need one each.
The solicitor they're talking about will be for you to take sis to court yourself. Either to contest the will and try to get some of the money for yourself or to bring about a private civil prosecution (as opposed to a criminal one) for the fraud of abusing her POA and stealing from DM.
You have a greater chance of winning with a civil suit than a criminal one because it goes on "balance of probability", not "beyond reasonable doubt", meaning you can win with less evidence/proof. But you can't insist on a criminal prosecution, even police don't decide that, it's for crown prosecution service to decide. You don't win a criminal conviction and resulting punishment for your sister with a civil suit, all you can win is some form of compensation, so basically no different in outcome to contesting the will and in either case if you lose, you'll be left with the solicitor fees. I can't see it's worth the hassle, especially when you're not bothered about the money anyway.
OPG telling you "get your own solicitor" is just another way of them saying that they're not going to do anything. They just don't word it like that because they know it's not what you want to hear and they CBA with you potentially kicking off at them down the phone like some people would. So they say "get your own solicitor" because it sounds like doing something effective, and is more likely to get you off the phone without an argument.
What they actually mean is "fuck off, we've closed the case, the victim is deceased and we're no longer interested" because their job is protecting the victim, not chasing criminals.
The police could chase her but it sounds like they've put it down to a family feud and would only proceed with a criminal investigation if OPG had proved sis had behaved unlawfully in her use of the POA. You can ask the police again but they've already told you they're not interested, basically. It's not like sis is a master criminal defrauding people left right and centre, so they've decided it's not in the public's interests to waste manpower determining whether she's done something wrong or whether you're just jealous you got cut out of the will.
The whole situation sucks. I see that. The care home inform NOK about the death, not all family members, because you're right they don't care about you, you weren't their client your mum was. And sis was the named NOK on their records. You need to speak to a counsellor to process your feelings if you're struggling, not waste your time and energy and drag down your mental health even more by trying to get someone official to do something about your sister. It's the equivalent of banging your head on a wall.
At least it's over and you can get on with grieving your DM and the relationship you didn't have with her. You're at a crossroads you can either destroy yourself fretting over all this or you can start to recover. I hope you start feeling a bit more at peace, in time 💐