I agree it would be more likely you could visit for up to 6 months a year. Unless they are rolling in money and can bankroll you, it's very difficult to get permanent residency in the states.
Another consideration is the location. Larger city? More rural? Transportation is a huge huge issue. Depending on your age and level of comfort in driving over there, it could affect you in a few years. Public transport is pretty much non-existent except in large cities. If you get settled over there and then reach a point where you can't drive for one reason or another, that's going to have a huge impact on your life, especially if you are in a rural area.
Health insurance is a huge expense, and you need to look at not only the actual cost of the insurance, but the level of coverage that insurance will get you. Even very good coverage can leave you with co-pays of up to £25 just to be seen in A&E (ER), as well as co-pays for any medication. If you have (or develop) any medical conditions that require ongoing medication, that's going to be an expense to keep in mind.
Also important to check things like life insurance, to see if moving abroad changes the terms. I have seen this happen, so it's worth checking into. If you're paying a lower rate now due to locking it in while younger, you may be surprised at the higher rate you might have to pay if starting anew in another country.
Also a consideration is retirement homes/nursing home care for when you're older. I have an elderly relative in the states, and she's been paying on a policy that covers care for her in a nursing home when she reaches that point, but she's been paying on it for years. You've not been there for years to build up social security and it's unlikely you'd get medicare benefits (or whatever it's called now), so how would you cover this?
Is she looking for child care for her children and plans on utilising you for this? If so, what happens when you can no longer do child care for her or if you simply don't wish to do it at all. Will they withdraw financial support or decide they don't want to "sponsor" you over there? What are the restrictions you need to follow? Is this considered work and will it violate any laws if you are "working" in this way? Again, I'd get them to spell out exactly what it is they are expecting and then do some research, as it would be awful for you to make the big move and then find out you're in a bad place over there due to circumstances.
There are other options - frequent visits of long duration, inviting the grandchildren over for visits during holidays or the long (3 months!) summer holidays. If it's just for the grandchildren's benefit, consider also that as they get older and go off to university, how much time will they be there in reality, while they're off carving out their own lives?