Another factor is that the German government tried to keep infection rates low at all times, leading to very little natural immunity in the population compared to other countries.
Beside that we can vaccinate a multiple of the number of newly infected people per day just fine. It's been about people being unwilling to get the shot, not availability of vaccines for quite a while now. Example: one day of vaccinations like yesterday (~1 Mio) [0] equals two weeks of new infections at levels like today (>73000).
[1] https://experience.arcgis.com/experience/478220a4c454480e823...