Flu's not caused by a coronavirus mum. Some common colds (and other respiratory diseases) are but it's worth rembering there is, as yet, no vaccine for any of the coronaviruses that infect humans. A flu jab will offer no protection against catching CV-19 at all. The idea of a wider flu vaccination effort is to help free up NHS resources to deal with an expected CV surge during the flu season and try to minimise the number of folk getting a double whammy of flu and CV at the same time.
The way that I have heard it explained agrees 100% with what you said. It is all about staying as healthy as possible so that flu patients and COVID patients are not competing for resources. Also, at least some of the symptoms overlap, if you are sick with the flu, you might be more inclined to need testing to be certain it is not COVID. And yes - the double whammy option is possible (AFAIK), and would be very bad.
The beauty of observing social distancing, wearing a mask, hand washing, etc. is that you are less likely to transmit (or conversely become infected by) either virus.