That's disgraceful! Your health Centre should know how many eligible patients it has for the flu vaccine. If they don't, they need someone to sort out their database. You should be able to get the jab at a time convenient to you, not necessarily Saturday morning. I have a younger asthmatic friend, who works Saturday Mornings, who is having to push for this service, but it's what she was advised to do by a practice nurse from another Health Centre. There'd be a much bigger problem if all the diabetics, asthmatics, very elderly folk and heart patients got flu and a proportion of them died ( which they would)
The 1918 Spanish Flu killed more people than WW1. Many of those were youg and healthy.
We have not many more anti-viral medicines now than then.