...Gefreiter was the only rank to which an enlisted soldier could be promoted.
The term "enlisted" is somewhat confusing here, as it translates poorly to the system of the German army.
In the Anglo-American system, you have two "classes": enlisted, and commissioned officers.
In the German system, you get three "classes": Privates, NCO's ("Unteroffiziere"), and officers.
If all you do is serving your conscription term (as the German army was a conscript army), you'll receive your basic training, then serve as a Private (Soldat / Gefreiter), and that's it. Insofar the Wiki page is correct -- you don't get promoted from Gefreiter to Unteroffizier.
So where do the NCO's come from? Easy: If you voluntarily extend your term, you'll receive your basic training, then receive NCO training, and rise to NCO rank (with accordingly better pay).
(This is, by the way, still the case in the German army today, even as conscription is suspended at this time. If you want to become Unteroffizier or higher in rank, you have to volunteer for a longer time of service and go through more extensive training.)
Of course, while the above was "the rule", there were always exceptions. A promising Private could be asked to attend NCO or even officer's training. Privates that had proven themselves in action could receive a battlefield promotion, although I would expect them to be sent back to receive the appropriate training at the next opportunity.
I was unable to come up with statistics on how often these things happened.
The best I could come up with during an ad-hoc web research was this paper (on page 21 ff.) talking about how, after WWI, the number of volunteers for officer's training did not meet the requirements. (The prestige of being a military officer had suffered badly, and a future in the shrinking military was an uncertain prospect in the early Weimar Republic.) So they filled vacancies in officer's training with promising NCO's and Privates, to the point where, in 1928, 3.5% of the officers corps (117 individuals) were former NCO's.
This is about NCO's reallocated to officer's training, not about Privates / Gefreite getting battlefield promotions to NCO rank. But I think it gives a general hallmark on both the existence and relative rarity of cross-career "jumping".