Is someone else’s surgery a legally permitted reason for international travel from the UK?

Your need to see a person before a surgery, or to bring some people to see that person, is not a medical emergency. Her need for the surgery, may be, but it is not her who wants to travel. I doubt you will get an exception.

Where you might have a chance is, it says on your link that you can travel for the same reasons you can leave home, and you can leave home

to visit someone who is dying or to visit someone receiving treatment in a hospital, hospice or care home, or to accompany a family member or friend to a medical appointment

Now, it also says you can leave home to exercise, and I’m sure you couldn’t get permission to fly to another country to exercise, so this isn’t a guarantee of success, especially since this “major surgery” is one that can be postponed a few times already.

I can’t see a way for you to apply for permission to travel; it looks like you just do it and perhaps argue with someone at the airport about it? This looks too risky to me.

My personal definition of emergency travel (under which I did travel internationally in November) is: does any cost, quarantine or other issue mean I won’t go, or will I go no matter what? I will go no matter what, then I go. In my case this involved two separate two week quarantines, one of which cost me over $5000, and Covid testing. Didn’t matter, I had to go. If your need is not that urgent, then you know you should not go.