make the best of every chance?

Your teacher could be right, Adverb, or she/he may be wrong.

There's nothing grammatic or semantic that would prevent you from saying what you've suggested.

Here are some exact phrase Googles. Maybe they will help, or they might get you in dutch with your teacher.

best of ...

Results 1 - 10 of about 1,740,000 English pages for "best of every opportunity".

Results 1 - 10 of about 921,000 English pages for "best of every chance".

most of ...

Results 1 - 10 of about 1,650,000 English pages for "most of every chance".

Results 1 - 10 of about 4,330,000 English pages for "most of every opportunity".

That's because one can create an opportunity but cannot create a chance..