To formally answer your question "Is controlling variables important in deducing a statistically significant correlation?", my answer would be: Yes, it's important, but the research question, setting and goal defines the exact reason (and importance) of controlling variables.
Before I explain, let's set a couples definitions. In my understanding a 'controlling' variable is an independent variable or predictor (so not the outcome of interest for your study), which is not the predictor of interest.
Second, let's call the predictor of interest the determinant, and let's shorten outcome of interest to outcome.
Further, one small disclaimer: my background is in biomedical research, so my explanation might contain some insights based on applied research, instead of fundamental.
Now (this is according to the school of thought I was taught) there are several types of 'biological' studies possible, but for this question the most interesting are (1) those studying causal links, and (2) those which goal is to predict the occurrence of the outcome. Let's call the former etiological studies and the latter prognostic studies (although for the latter other types can be included as well).
In etiological research we try to find out whether a certain determinant causes the outcome. In this type of research it is important to be sure there is not something else, which seems to occur with the determinant simultaneously, which is actually the cause of the outcome. In epidemiological research such a 'controlling variable' is called a 'confounder'. Do note, a confounder is not only something which also causes the outcome, but also something which is linked to the occurrence of the determinant:
When modelling this statistically, you include the determinant and the outcome, and then add all possible confounders in one go. Usually the researchers think of possible confounding factors based on prior knowledge, logical thinking, or because they are 'usual suspects' (age is often included as a surrogate for a whole lot of biological processes we can't measure or do not yet know of). This process is then interpreted in terms of 'crude effect', the effect without correction for confounding; and 'adjusted effect', the effect after correction for confounding. It is essential to understand that it is almost never possible to be completely rid of the effects of all possible confounders. This is called residual confounding. Some confounding information might simply not have been measured, while confounders which were measured might have been measured too crudely (think of age in categories instead of a number). This is were biological research differs from physics: AFAIK in physics we try to find 'laws of nature' (e.g. the law of gravity), and do not readily accept residual confounding. In biological research, however, we study associations conditional on certain conditions while accepting residual confounding (which the researcher often has to explain in the discussion). The only exception to this distinction are biological studies so fundamental, that they are essentially studying a 'law of nature' (e.g. does smoking cause a specific type of cancer vs. how do certain proteins affect cell physiology).
Secondly, there are prognostic, or prediction, studies. In these studies the controlling variables are not confounders, as the interest lies not with the causal association between the determinant and outcome, but simply with its predictive value (i.e. the strength or significance of the association). However, there might be overlapping information between the determinant and some other controlling variables.
Imagine you'd want to predict someone's risk of cardiovascular disease for some years to come, and you're interested in the predictive value of a certain blood panel. The thing is, a blood panel requires personnel to take and analyse the blood, and it might cause the patient harm. So, if age, gender and smoking habits (all obtainable simply through asking) can cover the predictive value of your blood panel, you might not need it. As you might understand, in this case you do not need theories about the controlling variables affecting the occurrence of the determinant or the outcome. As long as you think there is predictive information overlapping with the determinant, adding such variables to your model is indicated. Moreover, in the development of clinical prediction models, the ease of obtaining specific data is often considered more important than (a small) increase of model performance.
Puzzling the pieces together is indeed what is required for biological studies. However, more often than not, setting up these studies should be started based on 'non-statistical' reasoning (which variables are possible confounders; which variables are likely to predict my outcome and available in the study population), and not on statistical modelbuilding techniques. You could say that reasons for using specific controlling variables are therefore the best way for researchers to show the extent of their theoretical background in their analyses. But that this also shows that some information is 'uncontrollable'. Especially taking into account this latter statement, and (finally answering your question) looking at significance levels, or the size of the correlation or difference is definitely not the only thing you should do. Focussing on the reasoning for building specific models, which include and leave out certain (controlling) variables, is at least as, and probably even more, important as the final result.