Look at the steep decline in the COVID daily case rate. Vaccines WORK because they create herd immunity. You vaccinate the elderly population and to no one's surprise, the death rate begins to decline. If you look at the US chart, you see the same steep decline. From what I've read, Texas is now fully open in the US meaning that no more mask mandates and businesses can operate at full capacity. Perhaps we are rounding the corner of the COVID pandemic after all.
Vaccination definitely had a big role in reducing the fatality rates in countries such as United Kingdom and Israel. But that may not be the only factor. Currently the United Kingdom is experiencing the third wave. Before this, there was the first wave (April-May 2020) and the second wave (October-November). And in both cases, the peak lasted for 30 to 45 days. For the third wave also, the peak levels were reported only for around one month or so. So along with the vaccination, the natural progress of the pandemic may have also played its part.
And it needs to be seen how effective the vaccination is against some of the more resistant strains.