In the given example, it's not the demand for robust proof that is unreasonable, but what is inferred from the lack of its fulfillment. The idea that the lack of a robust study proves the opposite is an argument from ignorance.
The number of sources has no bearing on logical inference (cf. appeal to popularity). A single reference, if its premises are accepted, it's data reliable and unbiased, it makes a valid argument and its conclusion is sound, is sufficient. Conversely, a million references that each fail to prove the conclusion individually, may still fail to prove it as a group (though that number may ultimately wear down and convince any human). However, if a number of studies provide different data points, a meta-study can infer new conclusions. Take care here to distinguish logical inference from statistical significance.
While robust sources are desirable, they're not always available. Sometimes the studies just haven't been done yet, or it may not be possible (for physical, practical, ethical or other reasons) to achieve the level of robustness desired. That doesn't invalidate any argument or prove the opposite, it just leaves the audience with more space for doubt. An imperfect study can nevertheless be convincing if it can be ascertained that any flaws or biases were insufficient to skew the conclusion.
In some cases, absence of evidence can be taken as evidence of absence, for example when the claim is dependent on or predicts an effect that should be observable. However, we need to be careful, and like with positive claims, demand a valid argument to ensure the conclusion follows from what has (or hasn't) been observed.