What is the difference in nuance between 지루한 건 and 지루한 걸?

고쳐 주시는 건 (고쳐주시는 건) and 지루한 건 are correct.

The last two sentences imply (or indirectly say) that the speaker cannot accept something because it is bad, wrong, or disappointing; the speaker is making implicit comparisons/contrasts. For such implication, no one will use 고쳐 주시는 걸 or 고쳐 주시는 것이 and 지루한 걸 instead; the markers for comparison are 은 and 는. As you well know, 건 is the shortened form of 것은.

고쳐 주시는 건 감사한데 is expressing both the speaker's gratefulness for the listener's correction and the speaker's dissatisfaction about something (else); 대부분 지루한 건 알겠는데 is expressing the speaker's acknowledgment of the boring contents and unacceptability of something else at the same time. Their following clauses are explaining the speaker's dissatisfaction and unacceptability rather than directly mentioning the speaker's wish.