Justifying asymptotic worst-case analysis to scientists
My personal (and biased) take is that asymptotic worst-case analysis is a historical stepping stone to more practically useful kinds of analysis.  It therefore seems hard to justify to practitioners.

Proving bounds for the worst case is often easier than proving bounds for even "nice" definitions of average case.  Asymptotic analysis is also often much easier than proving reasonably tight bounds.  Worst-case asymptotic analysis is therefore a great place to start.

The work of Daniel Spielman and Shanghua Teng on smoothed analysis of Simplex seems to me a harbinger of what can happen when we start to gain a better understanding of the shape of a problem: tackling the worst-case first enables a more nuanced understanding to be developed.  Further, as Aaron Roth suggested in the comments, if the "usual" behaviour of a system is significantly different from its worst case, then the system is not yet completely specified and more work is needed to improve the model.  So going beyond worst-case generally seems important as a long-term goal.

As far as asymptotic analysis is concerned, it usually serves to keep a long and messy proof clear of distracting details.  Unfortunately there currently does not seem to be a way to reward the tedious work of filling in the details to obtain the actual constants, so that seldom seems to get done.  (Page limits also work against this.)  Careful analysis of the details of an asymptotic bound has led to actual algorithms, with good bounds on the constants, so I personally would like to see more of this kind of work.  Perhaps if more proofs were formalised using proof assistant systems, then the constants could be estimated with less additional effort.  (Or bounds on the constants, along the lines of Gowers' bound for the Szemerédi Regularity Lemma, might become more routine.)  There are also ways to prove lower bounds that are free of constants, by using more explicit machine models (such as deterministic finite-state automata).  However, such (near-)exact lower bounds for more general models of computation may require a lot of work, or be out of reach altogether.  This seems to have been pursued in ~1958–73 during the first heyday of automata theory, but as far as I can tell has since largely been left alone.

In short: in future, I think we'll see worst-case analysis as just a first step, and I hope we'll see expressions like $O$&#42;$(n^k)$ with galactic hidden constants as just the first step.  I find it difficult to justify the current focus on worst-case asymptotic analysis to practitioners, but perhaps it is useful to see this as the beginning steps of a longer journey.