Why "Throttle...Open 1/2 travel" on engine troubleshoot checklist?

Revised following update to OP:

If the engine quits and you're gliding along with it windmilling and the throttle pulled back to idle, if it comes to life at some point, you may not even be aware of it because you can barely tell the difference between off and idling when it's windmilling in a glide. On the other hand, pushing it full forward isn't such a good idea either.

You want it set to a power setting that makes it obvious it's come back to life, but you don't want to set it to full throttle because you don't want it going full bore instantly, which is generally not helpful to the engine because the oil pump may only be making 20 or 30 psi while windmilling, or none at all if it stops windmilling, and full power before the oil pressure can catch up is murder on the plain bearings of the crank journals that depend on oil pressure to keep crank journal and bearing separated from physical contact.

So half throttle makes a nice compromise.