I would like to add, that the power argument can still be used if you have a significant result.
Think about $200$ small studies which are underpowered due to their small sample size, e.g. the power in all of these studies is $40\%$. Let's further say that one half of these studies has bad luck, and actually the null hypothesis is true, i.e. there is no effect in reality. The other half is lucky and the alternative hypothesis is actually true. Now, using the classical threshold for significance $\alpha=0.05$, among the $100$ studies for which the null hypothesis is true, there will be $5\%$ of them which have a significant p-value, so we expect $5$ studies with significant finding (and only those get published in this simplified world). For the other $100$ studies, where the alternative hypothesis is true, we will expect $40$ significant findings. Hence, we have $45$ significant findings in total, from which $5$ are false-positives, so there is $5/45=11.11\%$ probability that a significant finding is a false-positive.
Now let's repeat the same thought experiment for the scenario of a world with only well-powered studies, e.g. power is $80\%$. From the studies with bad luck, there will be again $5\%$ that have a significant finding. But from the studies with an actual effect, now $80\%$ will have a significant finding, so we expect $80$ studies with significant p-value. Which makes it $5/85=5.88\%$ false-positives.
In conclusion: significant results of poorly powered studies are less reliable. However, the whole argument concerns sample size only indirectly, because sample size is related to power. A study can be well powered even with extremely small samples, and then I do not see any issue. To be valid, the full sample size argumentation just needs to be stated clearly before conduct of the study.
But in practice (at least from what I see daily), many small studies did not do a proper sample size calculation. To be honest, large studies don't do so much better. And many studies perform extensive multiple testing which increases type I error to a large extent, so the numbers that I have used in the example above are probably even more extreme in reality.