A smaller sample size is not better.
A small sample size needs a more significant* result if you want to draw a conclusion from it.
Let's consider some results and their interpretation:
If your drug cures 30% of 10 people, the percentage of the general population cured could be anywhere between around 0% and 65% of people.
If your drug cures 30% of 10000 people, you can be quite sure it actually cures around 30% of people (more specifically, between 29% and 31% of people).
If your drug cures 100% of 10 people, you can be quite sure it would cure around at least 60% of people.
If your drug cures 100% of 10000 people, you can be quite sure it actually cures around 100% of people.
Note: the above probably misses a few details about control groups, side effects, hypothesis testing, etc. It's just meant to give a basic idea of what the numbers might look like.
Now a one-line conclusion of a study could be "the drug likely cures some percentage of people" or "we don't know whether it cures anyone".
A 10000-person study is going to end up saying "the drug likely cures some percentage of people" more often, even if the percentage is really tiny. A 10-person study will end up saying "we don't know whether it cures anyone" more often.
When the 10-person study does end up saying "the drug likely cures some percentage of people", the percentage will generally be larger.
When a 10000-person study says "we don't know whether it cures anyone", we can be pretty sure that it cures between 0% and a very, very tiny percentage of the population. Whereas with a 10-person study with the same conclusion it could still cure a fairly large percentage. We just don't know yet.
But the results themselves are not more significant.
Note that above I didn't say "the results are more significant", but rather that you need more significant results. And I'm differentiating the results from the conclusion.
The quote (without context) seems to imply a smaller sample provides a more useful result, when this is blatantly false. This may not be what the author actually meant, but that's how I read it.
The results from a large study allows us to be more sure how effective something actually is, which is always more useful.
The only thing that would be more significant would be a positive conclusion ("the drug likely works"), but taking one look at the actual percentages would still give you a lot more information for the large study.
The only way in which a smaller sample would provide a more useful result is when people who don't know what they're doing misinterpret or misrepresent the result (by e.g. saying "the drug works" without also noting that it actually only works 1% of the time). This admittedly might happen a whole lot more often than it should in today's world with the media and social media.
What about bias?
If you have a very small sample size, you're much more likely to not have a sample that's proportional to what the actual population looks like, and you might even miss out on some demographic altogether.
In medicine there are many variables that could contribute to or alter the effects something has, so having an accurate representation of the population is quite important.
If your data is too biased, your results would not be particularly useful.
A bigger sample size doesn't automatically fix it, but does make it easier to avoid.
*: this answer uses "significant" to mean "practically significant" not "statistically significant". As in "something that actually matters to the general public".
Results from larger samples would generally be more statistically significant, as in it's something we can be more sure about.