I'm a PhD student in Europe, currently in my 3rd year of pure mathematics. It was supposed to be the final year of my studies before finding a position either in industry or in academy. However, the initial question my poisonous advisor has given me appeared to be a bit too hard (or as he formulated it "I appeared to be too weak") so he asked me to extend the duration of my studies for another year. As the university is very unhappy with people doing so, they force me to take 200 hours of teaching in this year (instead of barely 60). However, the university also offers people in their graduate studies to take a gap year in which for 6 months they work in machine learning startups without getting paid, and then 6 months in a startup in USA or Europe. I have quite solid programming background from my BSc and high school and I like equally pure mathematics and AI-related domains. Needless to say that after this year I am expected to finish the thesis, taking the teaching-penalty year.
My advisor and the graduate school said they agree for me to take this gap year "due to low number of positions in academy" and also a professor of applied mathematics told me it can open me opportunities for a job in industry and also for a post-doctoral position in computer science suppose one of my works gets published.
However I'm a bit hesitant and I wonder whether it wouldn't be better to finish my studies next year and only then to start looking for a job (probably in industry but things can change) fully paid. Also, my advisor is the current head of the graduate school and he told me "nobody would cry if you don't come back, suppose you're offered a good job" which might open problems with him after this year. What would you do? Can you share your experiences from similar situations?