When should risk averse people take on risk they can't diversify?

Self-protection means taking action to lower the probability of loss, when there are limits to self-insurance and there is no commercial insurance offered.  The questions we want to get at here are:

(a)  Sometimes, does the individual desire to self-protect get in the way of taking on socially desirable risks?

(b)  What, if anything, can be done about this?

Here are a couple of examples to illustrate:

(1)  Grant System Leads Cancer Researchers to Play It Safe.  The big deal issue here is how to evaluate the next grant proposal written by the researcher.  The prior track record matters, a lot.  In other words, if under the previous grant the researcher could show that good results were obtained that serves as a strong credential for the next grant.  In turn, that encourages the research to be less ambitious in the previous grant proposal, so success is more likely.

(2)  Underdogs in sports competitions should be risk seeking, to maximize their chances of winning.  Favorites should be risk averse.   But many coaches, whether underdog or favorite, coach not to lose.  Can you provide a rationale for that?

What other forms of self-protection do we see in the workplace or in the classroom?