For the purposes of a pilot, regardless of when it would turn out to be broadcast, it is difficult to make a categorical case that he should have done other than what he did. The show had to hit the ground running and any equivocation regarding his orders that would lead to a significant screen time commitment to wonky dialogue and a delay in action was not possible. However, given the information that Spock was able to supply, up to and including the Valiant's captain apparently ordering his ship's destruction, related to an impact that seemed clear to have occurred to a crewman because of exposure to the Barrier, a nod to at least to some kind of probity, examination, and thought by Kirk, rather than just ignoring such warnings, seems justified.
Such qualities could have been shown in short order, by his ordering of probe(s) that might have held the promise of returning varied types of telemetry of use in making what, instead of what turned out to be a heedless, reckless, and bullheaded decision, one instead that displayed a modicum of discretion. Would such probes have returned any information that would turn out to definitively rule out the attempt? No, of course not. But to have included such reasonable measures could have been executed using but, perhaps a couple of minutes of elapsed time on the broadcast. Would the concern that the Valiant was recorded to have desperately sought about espers, brought Kirk to an intuitive sense that such crew so superiorly endowed should be removed from the ship and stay outside the Barrier in a shuttlecraft say, while the Enterprise moved forward? No, lest there essentially be no story. However even a mention of concern about this issue could have at least passed from Kirk's lips.
Of course the ship was going to enter the Barrier. It would take incontrovertible evidence of potentially existential dangers to not execute his orders. But do you really think that Kirk's unquestioning, unthinking, and ramrod straight reflexive behavior in doing so, spoke very highly about certain executive thought processes that one would expect to be displayed by someone possessing this level of responsibility, especially since the loss of life had to be considered more than a dubious proposition? Interestingly, and perhaps significantly, do you really believe that Pike would followed this same template in making his decision? I most categorically think not.
