Chi-square of Fisher exact test for large samples

Question

The usual argument behind choosing chi-square or Fisher exact 2x2 test is that the Fisher exact test answers exactly a wrong question, while chi-square test answers approximately a right question. E.g., see Chi-square test or Fisher's Exact test

The problem is that if we mix the two tests in one research, then the tests will answer different questions. The problem is more acute if that happens for one variable in different samples.

So my question is: what is worse: mixing answers to different questions in one research; or using Fisher exact test for all samples while there is an opportunity to use chi-square test for some samples?

Answer 1

The problem is that if we mix the two tests in one research, then the tests will answer different questions. The problem is more acute if that happens for one variable in different samples.

I cannot see that this is correct. Both tests are for a null of independence (or homogeneity). For large samples they should tend to give the same answer.