I don't think anyones being entitled in this scenario.
The colleague shouldn't be guilted into giving up her xmas day, which the OP gets. Unfortunately, OPs family issues are not the colleagues responsibility.
The only one being unreasonable is the manager.
I have terrible memories of being guilted into giving up my Christmas. I moved away from home, and was working shifts, it was my first year doing it and I really landed on my feet with my shifts, I had an early shift xmas eve, then a late the day after boxing day. So i had booked my train back to my family home (I didn't drive at the time) for Christmas eve and travelling back 27th.
A colleague begged and begged saying she really wanted Christmas with her child (she had a 8-11 shift. I initially said no but got some passive aggressive comments of people how I was being unreasonable, I ended up giving in.
There was no public transport that day so I ended up paying £40 for a taxi, then I remrmber walking home trying to hold back tears (took about 3 hours) my sister messaging me saying my mum had been crying all morning that i couldn't make it back. I checked my Facebook and the lady i swapped with was videoing herself pissed as a fart at 11am, with all her extended family and lots of kids around. She lived 5 minutes away so could have easily just got her shifts over and done with and had the majority of the day. I got home at about 3, had a family video call then went to bed and sobbed.
No matter the scenario I wouldn't swap in the future (luckily no longer an issue as I don't work shifts)
I know this isn't the same scenario, as this is about care needs. But Christmas is just as important for childless people and the colleague is not a bitch for prioritising herself.