There are two questions here, specific and general.
In general, the logic of significance tests certainly does include consideration of sample size.
In my favourite software I get (without Yates' correction)
observed frequencies from keyboard; expected frequencies equal
Pearson chi2(1) = 4.0000 Pr = 0.046
likelihood-ratio chi2(1) = 4.1860 Pr = 0.041
+-------------------------------------------+
| observed expected obs - exp Pearson |
|-------------------------------------------|
| 4 8.000 -4.000 -1.414 |
| 12 8.000 4.000 1.414 |
+-------------------------------------------+
observed frequencies from keyboard; expected frequencies equal
Pearson chi2(1) = 8.0000 Pr = 0.005
likelihood-ratio chi2(1) = 8.3720 Pr = 0.004
+-------------------------------------------+
| observed expected obs - exp Pearson |
|-------------------------------------------|
| 8 16.000 -8.000 -2.000 |
| 24 16.000 8.000 2.000 |
+-------------------------------------------+
Specifically, it is hard to see why you describe the results as you do. Your output doesn't include observed significance levels at all, but either test yields significance at conventional level (all P-values are $\le$ 0.046, compared with the conventional level of 0.05 = 5%). The second has higher chi-square and thus for the same number of degrees of freedom a lower P-value (in ordinary language, a more significant result).
On the general point, take a different example: You're tossing a coin and get 6/10 heads. Is the coin fair (with probability 0.5 for heads, 0.5 for tails)? Do you regard possible proportions 60/100, 600/1000, ... 6 billion/10 billion, ... in exactly the same way? 6/10 could easily be a fluke, in ordinary language, but at some point as you get higher chi-square and lower P-values, you should change your mind.
The use of Yates' correction here is a side-issue except insofar as it affects threshold significance levels.