Another grand old hotel in NH is the Mountain View Grand Resort and Spa.
I stayed here 15 years ago and we loved it. We got a 2 room suite. One room for the kids. It had horseback riding then but it looks like they don't anymore. Probably the liabilty. It has golf and a farm, etc... Close to other activities like Santa's village, hiking, skiing, etc.... When we went they did a comedy night in the club. It is north of Cannon and has lovely views of the Franconia range and Mt Washington. We were there in the winter and it was if we had the whole hotel to our selves. I.E like in The Shining.
https://mountainviewgrand.com/
I don't know the current pricing of either but I've stayed at the Mt Washington hotel more recently and it was pretty pricey. The rooms we had weren't large or very modern. Of course you get what you pay for? No doubt the higher end rooms were much nicer. It is closer to Storyland & North Conway. They are both pretty self contained with plenty of amenities & activities.
Speaking of North Conway, there's a bazillion family resorts there. No doubt not too cheap either. A few have their own water parks. Much busier area with shopping & many restaurants, the ski museum, train rides & more. Some that I recall seeing are:
North Conway Grand Hotel
https://northconwaygrand.com/
Fox Ridge Resort
Red Jacket Mountain View - waterpark
https://redjacketresorts.com/resorts/red-jacket-mountain-view-resort
Hampton Inn & Suites - waterpark
White Mountain Hotel & Resort
https://whitemountainhotel.com/
I've only stayed at the last two in the winter, as empty nesters. I didn't even look at the pool area of the Hampton because it was vacation week. The elevator had plenty of wet kids.
The White Mountain is 3 miles out of the center right under the Ledges with beautiful views all around. The pool is on the smaller side. I believe the big draw here in the summer is golf, hiking and the Sunday brunch. We are usually checked out and skiing befote they were serving. :(
Then there's all the ski resort hotels.
The Grand Summits at Hunter, Mt Snow & Attitash. Jay Peak with their waterpark & golf course. The Mountain Club at Loon. I'm forgetting the rest....