I think it depends on the person's state of health. A 94 -yr-old lady, who I visit, had both 'flu and Covid booster jabs on the same day (in different arms). She's in a wheelchair anyway. She has been ill for over 3 weeks and is only now pulling round (a tough old girl!). It certainly wasn't a good idea for her. 2 other (older) people have said that they were poorly for about a week.
If anyone is worrying about having the pneumonia jab at the same time as a 'flu jab, then I can say that I was fine with that. Just an aside.