While, yes, you have to do do steps 1, 2, 3, and 5, my point was that #4, if your bag is checked through is not required. I've done it plenty of times. My origin airport checks it all the way to the destination. The bag tag has the final airport on the routing list. After customs, there's a no-human conveyor belt you drop your bag onto, which drops it back into the bag handling system. No need to re-check (if the origin Gate Agent did their job correctly).

And at least for LAX Bradley, there's a low-utilization TSA checkpoint set up for international arrivals (at least for daytime arrivals). It's "on the right" after you enter the greeting area past customs on the arrivals level. As mentioned before, T2-8 are all connected airside. In my experience, the lines are usually quite short compared to the main departures level checkpoints.