I don't remember a big push. That one study that showed deaths for 2020 were on track to be below the numbers for 2019 came out in October or November, but I'm seeing numbers (maybe fake) saying that we had about 500k more deaths in 2020. If fewer people were vaccinated for the flu in 2020, that could account for a huge spike in deaths.
Just a theory based on less than credible information.
There was a huge flu vaccine campaign in my region in Fall 2020. I'm talking radio, tv ads, couldn't walk into CVS without being accosted. They kept saying "you definitely don't want to get the flu AND covid at the same time."
According to the CDC's flu vaccination coverage data, 2019-2020 season was pretty much the same, even slightly higher for some groups, than the previous year. Seems like the same number of people that like to get the vaccine did so even with all the COVID craziness.