Until our Mongolian scholars check in, I’ll give my opinion. I think that many names of “foreign” people & places we’ve been using in English-language countries have been pronounced as early explorers & scholars understood them. In some cases, they probably didn’t hear what the “natives” were really saying. Also, languages change over time.
It’s a smaller world now. Even those of us who remember John Wayne playing Genghis Khan can just check out Youtube.
Not been to Mongolia, but throughout Central Asia it’s pronounced “Chinghis”, with a “ch” sound, which I think is also close to the Mongolian pronunciation.
I’d also suspect Bridget is right in that we may have ended up with “Genghis” because of early western visitors hearing it as that and putting it as such into writing (or first being written down that way by someone they then repeated the name to in that way)