恥 and 恥じる: evaluating shame and identifying its object
Compare judging concealment as shameful, embarrassment after a fall, and shame about a remark, preserving the condition on failure and the scope of negation.
A shameful evaluation and an embarrassing event
恥 (はじ) means shame or embarrassment. “失敗を隠すほうがかえって恥だと思う。” expresses the view that hiding a failure would be more shameful. The evaluated action is concealing the failure.
In “人前で転び、恥をかいた。”, the speaker fell in front of people and was embarrassed. 恥をかいた describes embarrassment at this event, without automatically admitting a moral wrongdoing.
Being ashamed of a remark and reading the condition
恥じる (はじる) means to be ashamed. “自分の軽率な発言を深く恥じている。” describes being deeply ashamed of one's own careless remark.
“努力した結果なら失敗を恥じる必要はない。” says there is no need to feel ashamed of failure when effort has been made. Keep the condition 努力した結果なら together with the negated necessity 必要はない.
Identify what is evaluated or felt shame over
恥だ evaluates hiding a failure as shameful; 恥をかいた reports embarrassment after a fall. 恥じている identifies the speaker's remark as the object of shame.
Do not restrict 恥 to public embarrassment or 恥じる to moral wrongdoing. The effort example also does not say every failure gives no reason for shame: read it within the stated condition.
What is negated in 努力した結果なら失敗を恥じる必要はない?
Show the answer and why
The need to feel ashamed of failure is negated under the condition that effort has been made. Do not remove that condition or read the sentence as saying no failure occurred.