No. They do not apply.

First of all, ANOVA does not apply at all. Because ANOVA is for interval scaled dependent variables. There is no interval scaled variable anywhere in your study description.

Chi square could be used possibly, but Chi square is useful most if both variables are categorical. Here, one of the variables is categorical, but the other variable is ordinal! It is better to treat an ordinal variable as ordinal, and not to treat an ordinal variable as categorical. Because, treating an ordinal variable as categorical does waste information.

Therefore, a test should be used which considers that one variable is categorical (binary, which means: with 2 groups), and the other variable is ordinal. Such a test which for the relationship between an ordinal variable (such as age group) and a binary variable (success yes/no) is the Mann-Whitney U-test.

By the way, you will absolutely not be able to "prove" that age is independent of performance. Maybe your test will fail to show a statistically significant relationship between age and performance. But with such a small sample size (n=29), your study is not sensitive enough to reliably rule out any relationship between age and performance.

With kind regards

K.