Here's the important thing to know: ANOVA is just a fancy way of saying "Doing an F-test on a linear regression."* (I would ban the word "ANOVA" if I could, to be honest.)

OK, so we can replace the word "ANOVA" with "Linear regression." Now the thing to understand is that the linear regression is Simpson's "paradox." Simpson's paradox is just the observation that a linear regression coefficient** can be negative even if the population correlation is positive, as long as the linear regression stratifies by (controls for) more than one variable.

I'd suggest taking a look at Richard McElreath's book Statistical Rethinking; I think the more intuitive approach he takes there might be helpful.

*It's not, strictly speaking, necessary to calculate the linear regression out if all you're interested in is p-values. However, the numbers you calculate at the end are identical and the two procedures are mathematically equivalent.

**Or another kind of analysis.

Here's the important thing to know: ANOVA is just a fancy way of saying "Doing an F-test on a linear regression."* (I would ban the word "ANOVA" if I could, to be honest.)

OK, so we can replace the word "ANOVA" with "Linear regression." The next thing to understand is that the linear regression is Simpson's "paradox." Simpson's paradox is just the observation that a linear regression coefficient** can be negative even if the total correlation in the population is positive, as long as the linear regression stratifies by (controls for) more than one variable.

Because of that, we can see that we don't want a methodology that's "immune" to Simpson's Paradox. If I add a control variable to a linear regression, I'm doing it because I want to know the partial correlation -- the correlation after adjusting for this variable -- not the total correlation. If you want the total correlation, calculate the total correlation, not an ANOVA or linear regression.

I'd suggest taking a look at Richard McElreath's book Statistical Rethinking; I think the more intuitive approach he takes there might be helpful.

*It's not, strictly speaking, necessary to calculate the linear regression out if all you're interested in is p-values. However, the numbers you calculate at the end are identical and the two procedures are mathematically equivalent.

**Or another kind of analysis.