尽くす and 費やす: effort, devotion, and time spent
Compare athletes' utmost effort and a teacher's devotion with three months spent on a model and a wish to avoid unnecessary work, preserving effort, duration, and intention.
尽くす: utmost effort and years of devotion
尽くす (つくす) means do one's utmost or devote oneself. 「選手たちは最後まで全力を尽くした。」 means “The athletes did their utmost to the end.” The athletes did their utmost to the end. 全力を尽くした gives the degree of effort and its persistence to the end, without reporting that they won.
「長年地域の教育に尽くす先生を皆が尊敬している。」 means “Everyone respects the teacher who has devoted many years to local education.” Everyone respects the teacher for years devoted to local education. 尽くす concerns sustained devotion here; retain the teacher as the person respected and local education as the focus of the work.
費やす: time until completion and work to avoid
費やす (ついやす) means to spend. 「模型の完成までに三か月を費やした。」 means “It took three months to complete the model.” Completing the model took three months. 三か月 is time spent, not three attempts or the duration of the athletes' competition.
「不要な作業に時間を費やすのは避けたい。」 means “I want to avoid spending time on unnecessary work.” The speaker wants to avoid spending time on unnecessary work. 避けたい expresses a wish, without establishing that such work has already been eliminated or that no time was spent on it.
Read effort, duration, and intention separately
With 尽くす, look at effort or devotion. With 費やす, look at time spent. Doing one's utmost and spending a long time do not guarantee the same successful outcome; retain only the results the examples state.
The athletes, the teacher, the three-month model, and the wish to avoid unnecessary work belong to separate examples. Effort does not establish victory, and a wish to avoid waste does not establish completed removal of the work.
What do the athletes doing their utmost (全力を尽くした) and the three months spent on the model (三か月を費やした) each tell you?
Show the answer and why
The first gives the athletes' utmost effort to the end; the second gives three months spent completing the model. Effort alone does not establish victory. The separate wish, 避けたい, to avoid unnecessary work also does not establish that the work has been eliminated.