'obstinately' is the proper word for expressing '굳이' I think. Not exactly, but almost same meaning it has!

I would say different words can be used for 굳이 depending on the usage. The general sense is of insisting on something that is not entirely necessary, or being too particular or obstinate in one's demand. - 굳이 그러시겠다면 = If you INSIST on doing so. - 굳이 그러실 필요 없어요 = You don't PARTICULARLY need to do it. - 그는 폭우 속에서도 굳이 떠났다 = He OBSTINATELY set out braving the torrential rain. - 어머니가 굳이 과외를 시키니까 = as mother (WENT OUT OF HER WAY TO send me / ADAMANTLY insisted on sending me) to cram schools. But all these are approximate translations. No English expression seems to have a very close meaning.

"to go the extra mile" would be the best fit in short to put some extra efforts for an arguably questionable output You often use this adverb when questioning the "bang for the buck" and/or the necessity.