Well, kind of. I wouldn't argue that he displayed a "captainly" mien from the outset. There was no question that he embodied important characteristics of a strong leader, being decisive, forceful, and extremely self-possessed.
However, I think that one can suggest that his decisions in this maiden adventure belied an attempt to unequivocally pronounce his bonafides as an experienced, thoughtful, and prudent holder of such an exalted command. Putting asides the obvious dictates of the premiere of this type of show to hit the ground running, we see someone with a momentous brief to carry out, one with a significance that doesn't need any further set-up for even a casual viewer to appreciate, and how does Kirk proceed? As if the mission description must have a descriptor somewhere, underlined and in bold, Beat the Clock!
Kirk is faced with a number of uncertainties that bear directly on how this venture into the unknown should be handled, presumably with at least a modicum of safety and responsibility. What was the nature of the damage to the Valiant? What are the physical qualities of how the Barrier is constituted that resulted in the Valiant barely being able to escape their encounter with it? What lies behind its crew's frantic effort to gather information on esper abilities and how is the agency of the Barrier connected to that desperate search? Further, did such a connection point to some exceptional outlier being present among the Valiant's complement or could such an unknown effect just as easily impact crewmembers on Enterprise? Finally, what possible threat could have apparently forced the command decision of self-immolation to be made and those dire warnings left for those who would follow?
These factors would seem to suggest a sober, cautious course of action to be questioned, reasoned out, and then implemented. Perhaps dispatch a probe first and from results gauged by its progress, extrapolate the potential impacts on Enterprise. I know, the show is only an hour long and as the first offering, its creator and the network heads likely didn't want the audience's first impression of these explorers to be of timorous, risk averse plodders. Still, brevity could have been served with a limited amount of dialogue that would have indicated that this captain, this Kirk, would have ordered some kind of deeper analysis and consultation amongst the undoubtedly disparate and superlative experts on all relevant disciplines to this matter that he had to choose from in his crew. Such limited exposition could have simply informed us that unfortunately these efforts were of limited utility, if any, leaving slight option as to the next step.
But no. Kirk's leap into the void was portrayed as little more than a seat of the pants affair, recklessly and rashly entered into with no space, seemingly, for contingencies or precautions. What, if they took longer before making the foray into the Barrier, the time allotted for the mission would be exceeded and the entire crew would have to be paid overtime?

Perhaps, at some level, we were being shown an unthinking overconfidence in how their vastly advanced technological advantages over their predecessors would bail them out of any challenging situation, however daunting and unknown. Or, maybe it was a statement, uttered sotto voce, that this strong willed leader who seemed to have no lack of certitude in his bearing and manner, was in fact a commander still in the early days of his tenancy of the chair and that one could see vulnerabilities in his judgements, ones that we would presumably watch being addressed as each upcoming trial would add to his seasoning.
At the end of the episode, Kirk feelingly says that his now deceased friend didn't ask for what happened to him. I've long thought that at some point down the road it would have been appropriate for an introspective Kirk to question and reflect on his culpability for what befell Mitchell. It would certainly have made for an interesting sidebar perhaps, as a window on his interior struggles and doubts that we sometimes saw enacted. Having read very little of the novels, I'm curious if this experience was ever referenced in such a way as a further illumination of Kirk's character.