I might be wrong, but they only have to take one if you are below a certain age _and_ below the border income (Beitragsbemessungsgrenze). Always be careful w.r.t. these kinds of decisions because they could cost you an awful amount of money later.
If you are above the border income, the private version is less expensive (I might also be wrong, checkout https://www.finanztip.de/pkv/pkv-basistarif/). It only really affects people below that income (way too many!).
It is the reverse, staying in the gov insurance costs you an awful lot of money upfront. It's the ethical but financially stupid thing.
True, the private insurance is typicall less expensive now (when I researched, it was 50%-60% of what I pay), but you have no guarantee that it will stay that way. They could just increase the fees for everybody, in the future, or just increase it with your age, as the health risks go up.