調べる and 発見する: investigation and discovery
Separate examining something from a discovery.
Looking into information
調べる (しらべる) means examining information or an object. 「電車の時刻を調べました」 says that the speaker checked train times. It is not limited to research revealing something new to the world.
「辞書で意味を調べます」 means looking up a meaning in a dictionary. The object is a meaning, so this is not a search for a person.
What was discovered
発見 (はっけん) means discovery. 「資料の間違いを発見しました」 says that the speaker discovered an error in the material. It may have happened during an investigation or by chance; the word does not specify how.
「偶然、新しい店を発見しました」 means discovering a new shop by chance. 発見 is not restricted to results following a planned investigation.
Investigation and outcome differ
「詳しく調べましたが、原因は発見できませんでした」 means that the speaker investigated thoroughly but could not discover the cause. Automatically translating 調べた as 'found out' would conflict with that unsuccessful outcome.
「調べれば分かるかもしれません」 says that investigating may reveal the answer. Rendering it as an established fact would lose both the condition and the uncertainty.
What is the sequence in 「発見した間違いを直しました」?
Show the answer and why
The error was discovered and then corrected. Discovery and correction are separate actions.