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"So I ended up booking outbound on Uniter and return on American."
You will certainly miss your flights if you show up at the United terminal at LAX. Your outbound is not on United.
If you mis-connect somewhere along the way, you do not get to "pick" other flights; the airline that delivered you late to the connecting point is responsible for re-routing you to your destination, which might involve going through completely different cities than what you have on your current itinerary. (You can suggest routings to the airline agent, but the airline ultimately decides how to re-route you.)