Reasons to be Consistent, Part III:
1) Defense: smiling, an estimator property is "asymptotic" when we don't have a clue as to when it will actually start to visibly affect the behavior and the results of an estimator. It may take a sample of immense size, it may take a few dozens of observations. So we want to have consistency in order to safeguard against being led ashtray, and without even knowing it. Since patients' health is at stake, I guess this alone should be an argument that health professionals would listen to.
A graphical exposition could posses two values, the true value (and the associated treatment of the patient), and the probability limit of the estimator (wrong value), and its associated treatment. If the treatment in the second case is different (and possibly irrelevant/detrimental) than the one under consistency, then you have the potential danger the users will bear (and impose on the patient) by using the inconsistent estimator.
2) In most cases, inconsistency also means the existence of bias even if large amounts of information gather (although strictly speaking, the concept of (un)biasedness at a limiting situation has more than one definitions). So you could fall back to something like "even if sample sizes are small and you don't think that inconsistency matters, as measurements accumulate, if you pool them, their average will also be wrong" -since the averaging operation is something that everybody feels familiar with. So inconsistency makes pooling of the obtained estimates, or of the data proper, misleading, something that sabotages any mid-term / long-term attempt to uncover the true situation.
Reasons to be Helpful, Part III:
Can you give them a positive result? Is there an alternative estimator that performs the same job, and is also consistent? And if yes, how it compares as regards finite-sample properties, like bias, variance, Mean Squared Error?
Reasons to Worry, Part III:
The real tough situation would be if a) there is no alternative or b) the alternative is consistent but it performs worse in finite sample properties. Here you enter into Risk and Decision Theory proper, in which case, @whuber should jump in and clear the fog.