There is a lot of overlap between rights/powers on death and right/powers on divorce. E.g. If husband gives away (by will on death) everything owned in his name, and every thing IS in his name, then the wife can still ask the courts to give her the same half of the estate that she would have got if she divorced. I am not a lawyer, but I know this is a problem that needs a lawyer's expertise. Don't skimp on costs at this point.
Any prior will becomes invalid on remarriage, so new ones are commonly signed at the church door. The two people should both do both together.
The link's response said clearly that you are obliged to carry out your mom's wishes. I am not sure that is correct. She made the decision to trust your best efforts when she gave you the POA. You must make decisions in HER best interests (not necessarily her wishes) though, not necessarily in the best interests of you kids.
I believe it is common with second marriages, that there be frank discussion about the parent's wishes to leave most of their estate to the kids, instead of leaving even half in the hands of the second spouse who could chose to leave it, in term, to his/her own children by prior marriage.
The will can have a clause saying "anyone who disputes this allocation will lose all rights to what is specified here". That can put the risk of not-winning on the surviving spouse.
The will may put the assets into a testimentary trust (created within the will) trusteed by an independant trust company. The specifications can (eg) be that the spouse has rights to continue living in the house and receive div/interest from any portfolio, with the capital reverting to the children on spouse's death.
Testimentary trusts are great vehicles. They are taxed just like another individual, with a basic amount untaxed, and incremental income taxed at rising rates. But the beneficiaries have essentially NO RECOURSE if they disagree with the trustee. So best to specify DUAL trustees with a family representative along with the professional.
The will may specify that certain assets go straight to the spouse "with the wishes that the spouse will leave any residual value to the other spouse's kids at the later death". But this cannot be enforced in any way. And believe me, people's attitudes about what they 'deserve' will not be what you think.
Coming back here later I see I have talked about the will, but not really the agreement. There may well be an idea that you can get around the bad feeling of the agreement by putting all the stuff in a will, but I would not rely on that. Much better to have the same terms in both documents.
P.S. watch how you use terms like 'already'. Even the kids don't know what goes inside a marriage. Don't judge.