Intermittent start ~2years, 2015 3.8L Genesis Coupe Hyundai

Over the last 2 years, my car will have random fits of not starting. not the battery. not the peddle (at least i dont think so, “press peddle” goes away when it asks me to and i do).
I press ignition and nothing happens. this video shows an instance of the issue.

Sometimes it starts right up, will be fine for a couple days or even a week.
other times it takes 2-3 min, and sometimes 20min of frustrating button pushes and channeling inner chacra. It has always turned over for the last 2 years dealing with this.
Until today I tired for 40 min and gave up. wake up call to just get it done.
Looking around i see [this starter on rock auto, 11 teeth
and [this one also, but 13 teeth
and chat gpt says 12 teeth! What is the correct starter i need for my car, assuming that is the issue?
Its getting warmer outside now and I’m guessing after all this time the temperature change (im in MD) is causing bushings to expand ever so slightly making it even harder… that’s my best educated guess not knowing much about cars.

Probably unrelated, but there is a screw in my oil pan, any idea what it is and how critical it is?

it falls back in when i pull off drain plug, but ive been just leaving it attached and its still there when i change oil again