'Making the call' means making a final decision for one option over another. The origin of the phrase seems to originate from US sports, in the context of a decision about whether a foul has been committed or a ball was out. It might or might not be related to 'calling' an outcome. Or it could plausibly be related to 'calling' in poker, which ends a round of betting and pushes forward the action.
Decisions are largely binary. They have to go one way or another. The very idea of facing a decision implies rivalry between your options in the sense that you can only take one course of action and not lots of them. But beliefs are continuous. You can assign some belief (in the form of probability) to a large range of possible facts, or hypotheses. A simple boundary between the role of the analyst and that of the decision-maker is that the former assigns probability to a range of possible facts, and the latter deploys resources in exactly one way. Decision-makers 'make the call'.
Should analysts 'make the call'? The best interpretation I can put on this idea is that it implies assigning all of your belief to a single hypothesis, or at least communicating to a decision-maker as though that's what you've done. Certainty undoubtedly sells, in the media as well as in business and government. But it is not a good strategy for making optimal decisions. Good decision-making is about finding value, not about taking the option which is most likely to produce a positive outcome.
As with all decision-making under uncertainty, it's exactly like betting on a horse race. A horse with a 10% chance of winning at odds of 20-1 is a profit-making bet, whereas a horse with a 60% chance of winning at odds of 3-1 on (i.e. 1-3) is a loss-making bet. An analyst 'making the call' would declare the first horse a loser and the second a winner, and the wrong bets would be made. Without the probabilities to hand, a gambler - or a strategic decision-maker - is not in a position to identify the optimal course of action. Analysts need to stick to probabilities and let decision-makers make the call.