The somewhat easier target, a little to the deontological left of your question ("why should"-> "why would"), sheds a little light on things. For almost all philosophers believe in something akin to the so called 'golden rule of ethics' (the form most germane here being: "All moral acts remain moral if committed by everyone"), and almost all that practise philosophy do so for some ethical consideration. The syllogism that ends, therefore, in almost all philosophers prescribing philosophy on ethical grounds is a moment's work to complete.
But now, how to skirt the guillotine and leap from is to ought? Perhaps one could avoid such considerations of ought altogether, observing that all people introspect
...and it is the work of Socrates & co to help them do what they seek to do already more fruitfully.
This feels a little unsatisfying, though: somewhat straw-mannish to claim there is nothing qualitatively different in the self-examinations of the Athenians and those of Socrates. At the risk of incorrectness, then: Mr. Hume- to the scaffold!
Building on the original dummy question ("Why would..."), I would like to reach for my meta-ethical breeze-block of choice, in R.M. Hare's universal prescriptivism, which ascribes the origin of moral injunctions as non-directed imperatives that emerge from a society ("Don't kill!"). In this framework, the morality of prescribing introspection is identical with the morality of introspection itself, which in turn stems (as articulated by the OP) from the imperative that as moral beings we must grapple with reality- something we would fail to do without introspection.
Again, though, (although my use of Hare is based purely on my thinking he happens to be right) I suspect people will think I'm cheating here. So perhaps, to be as general as possible, let's talk the social contracts.
In the end, it is a truism that groups with common goals and common values achieve those goals and actuate those values with greater efficacy than the leading brand. The impulse to spread a cultural more that is beneficial (or at least impossible to get rid of- once you introspect, aside from alcohol or a coathanger up your nostril, there's no going back!), is one that leads to a more cohesive, more productive society, more capable of modelling a universal in-group.
There is selfishness there too, certainly: If I never saw another TV talent show again, I would be a happier man. But, impractical (though possibly correct) calculations of the relative happiness of Socrates and pigs aside, the moral heart that drives evangelism of the examined life is, in one way or other, the picture of humanity as a shared endeavor that only introspective thought can show in its fullness.