I worked for the same organisation for many years, and there were various processes for sharing leave. None were ideal, but management did finally at least try to be fair.
The office was closed Bank Holidays, but the time between Christmas and New Year was always sought after, and to start with around 50% of staff were allowed off.
As a young childless person I was told I had to work the first year. For the next 6 years I ended up working with different women with children. For each of those years I was asked to work, with the promise of being off the following year. It never happened.
I had to make a stand and on year 8 I flatly refused. There was much emotional blackmail, requests to share the time off, etc but I stood firm. My own manager supported me, but a more senior one threatened disciplinary action. I had followed all proper procedures and applied for leave in writing, so had written rejections for 7 years. I threatened HR and ACAS.
I got the leave and for once was able to travel to see more distant relatives.
There was more fairness after that and whilst there were exceptions for special circumstances everyone was expected to cover their share.
I feel strongly that it is up to management to manage, not pick on the weakest target. They would have to make decisions if everyone in the team had children.
I read the other post about how hard it is to get childcare at Christmas, and am not unsympathetic, it’s sometimes a similar situation for adult carers, but if a manager ( not unreasonably) accepts that as a reason for allowing leave over and above the desired level, it is similarly up to them to fill the gap or accept lower staffing levels.
Filling the gap could be by various means such as using temporary staff, or by offering financial / time inducements to actual volunteers.