맞춰 발전시키다 is 맞추어서 발전시키다 if spelled out fully, so it is an example of the -아/어서 ending. When used about two actions (not a reason which is another common use), -아/어서 may indicate the first of two consecutive actions but sometimes it just explains the means or manner of the action that follows. For example, 커피에 설탕 넣어서 드세요? is more about how they take coffee (i.e. sweetened) than a separate action step before drinking it.
Likewise, 맞춰 in 맞춰 발전시킨 means a detail about developing it and not that they did the 맞춰 part first. It means developing it so that it blends in with the local ways, so 맞추다 and 발전시키다 happen at the same time.
I think the to-infinitive in English is somewhat similar to -아/어서, in that sometimes the infinitive part may be more distinctly separate from the main clause but sometimes it is not. Here "developed Confucianism from China to fit the situation" is a case of the latter, since you can't easily develop something first and then fit it to another thing later - the two must happen in parallel.
So both the original and the translation mean developing Confucianism in harmony with the local customs. Neither is saying you do adapting and developing in two separate stages. If you think of it this way, I think the translation is correct (although it does seem to emphasize the "fitting" part more than in the original, but it's not a huge difference).
