I'm an earlier-in-career software engineer, and I work in an extremely busy startup on a small team (<10 engineers). I do a lot of things, including:
However, at the end of the day, tickets need to get done and I've had some trouble with that. How can I get better at this? Do I just need to "put in the reps" when it comes to coding?
I empathize with this a lot having also worked at an early stage startup, and you are 100% correct in that you need to put in the reps with coding. The crucial thing to do now at this stage in your career is to build an extremely solid coding foundation, which you can't do if you're unable to focus. Here's what I recommend:
All that being said, it will be hard to do this in a very small startup like yours. Do what you can, and regardless of what happens, keep an eye out for gaps that can seriously damage the company. Even though your goal is to drill down into a specific code area, that doesn't mean you should step outside of that zone when times really call for it, especially in a startup.