Yes, it can be cold in Spain, but usually not super freezing with heavy snow like parts of the USA East coast! However, it is not really warm enough to swim in any beaches.
Actually, I do like your new itinerary--it is much more slow paced and relaxed. I think you have allowed sufficient time to really see some of the sites in a meaningful way. My only thought would be that if you have the energy, you should go directly from Madrid airport to the train station and straight to Sevilla. Yes, you will have jet lag, but you can check in right away to your hotel and have more nights there instead of having to check in and out of a hotel in Madrid for one night.
A caveat would be to do a deep dive to check opening and closing hours of sites during the holiday season, as often there are religious celebrations which may have some impact on accessibility, especially for cathedrals. You may need to buy some tickets in advance online, such as the Alhambra.
I am also going to recommend that you consider making meal reservations for the holidays, just because some restaurants may be closed. Your hotels should be able to assist with recommendations.
Your children are at a good age where they can certainly learn in advance about some of the things they will see. You want to make the trip fun for them, not just a history lesson (but be sure to see Columbus' tomb!). They do need to learn why Andalucía is full of Moorish architecture.
If possible, consider going to a flamenco show, and maybe even a soccer game! Make sure you do a tapas dining time with them, and don't forget to try the chocolate and churros. Some people like to do fun things like bicycle tours with kids. The children should definitely learn and practice some Spanish!
There is so much to see and do in Madrid that I do not know where to start for just a day, but if you like royalty, consider touring the Royal Palace, if you like art, consider visiting the Prado, and make sure to get to the Plaza Mayor.
Here is Spain's official tourism website:
https://www.spain.info/en/
Have a fantastic time planning, and happy holidays. ¡Buen viaje!