I saw this question on Strikerdafastest's wall and felt compelled to answer, and it ended up being one of my famous novel-length posts, but the topic ended up removed by the time I finished, so my work was lost to the ether. Therefore, I reanimate Reaper's Blog 4 Thinkin' to bring my wall of text to you.
To begin with, here are potential in-universe reasons:
- Striker Eureka is a very complicated machine. As any engineer or fan of Mythbusters will tell you, the more complex a system is, the more likely it is that there will be failures or other hiccups. On a machine designed for battle with monsters that are 20 stories tall, this will be especially true (imagine the G loads on all that delicate internal machinery!), even when things are going smoothly (for example, one of the movie's deleted scenes has Chuck noting after the battle with Mutavore that some of Striker's hydraulics are broken and must be replaced, even though Striker won without breaking a sweat). Of course, there's no doubt that Striker has redundancies (Slattern supposedly knocked out half its systems in the first attack, but Striker still kept fighting; and Gipsy has a history of fighting while missing limbs or having gaping holes in its head or chest), but any mechanical or electrical problem anywhere down the line (like from, say, minor electromagnetic pulses coming off of Leatherback while he charges the big one) could easily slow down the deployment of a weapon system. In support of this is the fact that the time it takes for Striker's AKM launcher and Gipsy's plasma cannons to prepare to fire vary.
- Chuck and his father do not have a great relationship during the Battle of Hong Kong. As we know (and saw in an extreme form during the Gipsy Danger Drift Test), any (momentary or otherwise) problems in the Drift can have significant effects on a team's ability to fight. A temporary disagreement or distraction could lead to hesitation.
- During the only time we see the AKM launcher successfully used, the K-Stunner missiles are not depicted as being particularly accurate (they hit Mutavore all over the place, instead of being focused on one spot). It has been speculated that the missiles do not have any anti-(kaiju)blueshed properties and were therefore intentionally fired that way to minimize penetration and kill Mutavore solely through impact force. If one takes the stance that this is not the case and that they do in fact require extra time to aim properly, one could then assume that the reason Striker takes extra time before firing is to ensure that the aim is good (not only does Otachi have a higher speed than Mutavore and no buildings to get in the way if she attempts to dodge, but Striker is facing an urban area without actually being in one, thereby making blueshed somewhat less of an issue and missed shots a much bigger one). In fact, one can take this interpretation even if they believe that Striker's missiles don't have accuracy issues.
However, practical issues aside, there is an underlying issue with the question of why Striker Eureka didn't fire its missiles sooner and finish off Otachi. That issue is that it carries the erroneous assumption that Striker would have been allowed to defeat Otachi at all.
When you get right down to it, just about everything happens for the sake of the plot. No matter how quickly Striker Eureka had attempted to fire its missiles, it wouldn't have been quick enough (or they would have missed, or been ineffective, etc). Striker was always going to lose that fight, because it wasn't his narrative role to win there.
Really, if we're going to take out a stopwatch for this, even if we assume that the battle between Striker and Otachi started the instant Otachi left Cherno Alpha to Leatherback, the fact is that Leatherback finished Cherno in 35 seconds, while Striker spent 48 beating Otachi senseless and preparing its missiles. With 13 seconds (or more) during which Leatherback could have ambushed them, the Hansens shouldn't have even gotten as close to shooting as they did.
So what happened? The narrative happened.
The purpose of the Battle of Hong Kong (part I) was to create a situation desperate enough that Stacker would have to let Mako and Raleigh take the Gipsy Danger into battle, save the day, and prove themselves.
To make this work, the two Kaiju (and this is also why there were two instead of just one) had to be proven as genuine threats. This required the sacrifice of redshirts (technically mauveshirts). This was why Cherno and Typhoon existed (besides pandering and international symbolism). They and their crews were created, introduced, and used to fill out backgrounds so that when the time came, they could be sacrificed to show how dangerous Otachi and Leatherback were.
Striker Eureka, on the other hand, was a tougher nut to crack. It had to be out there, of course (otherwise it would have sent a mixed message about how big a threat the Double Event really was). And it couldn't defeat the Kaiju (this was Gipsy's time to shine). However, it needed to be around for the final battle, and it couldn't be defeated in a straight-up slugfest (the entire reason Striker was depicted in the Mutavore and Otachi fights as so physically dominant was to get across to us the sheer might of Slattern when it began beating Striker at its own game). And in a stroke of brilliance, the solution del Toro and Beacham came up with was to have Striker be disabled by something completely out of left field that had nothing to do with physical might and left no real damage (and happening just as Beacham and del Toro were teasing us with the idea of Striker winning, to boot). Striker's reputation as a physical powerhouse was left untarnished to better serve the drama of the final battle, Herc was able to be injured without killing him or injuring Chuck as well (Convenient or intended? I think we all know the answer by now) so that he and Stacker could switch places, Striker was able to participate in Operation: Pitfall, and the stage was set for Gipsy to save the day in most viewers' favorite part of the movie.
The point I'm getting at here is that in the end, everything moves at the speed of plot.