when starting the engine, it sounds like the sylinoid (?) sticks after the car sits idle for more than 30 minutes-ish.

What is your question?

if I shut it down, it cranks right back with no sticking noise. Already replaced starter, Starter relay, and ignition switch.

What makes this problem better or worse?

Its worse when absolutely cold, like overnight

How long has your 2008 Toyota Sienna had this problem?

last few months