You may not be wrong middle:https://www.statnews.com/2015/11/11/flu-shots-reduce-effectiveness/
It's also true that the flu vaccines are less effective in older populations.
As for the dog, she has to go to daycare, no way around that with her separation anxiety. Luckily they accepted the waiver from the vet.
They do say the canine vax will decrease the amount of viral shedding that a dog does after they come in contact with the actual virus, but I couldn't find a list of ingredients, including adjuvant.