Is it reasonable on a conceptual level to say that the Chi-square test is equivalent to performing a bunch of Z-tests at once, and that the F-test is equivalent to performing a bunch of t-tests at once?
[Note: I don't have very much formal statistical background]
According to this Stat Exchange post, a "z-test" and Chi-square can be compared to one another given they both focus on the same relationships, but you can't repeatedly run a Chi-square over and over without potentially having false-positives (or results the don't "preserve the alpha" to quote the link).
Chi-square tests analyze what you observe with what you expect, and quantify that relationship.
Z-tests are how well a test statistic (under a null hypothesis) matches an approximation to the Bell curve, or normal distribution.
in Chi-square, you are looking at how well your results matched your expectations, and with z-tests you are analyzing how well your recorded data can be approximated with a normal distribution.