From my daughter the bird nerd! :
The closed eyes definitely indicate it's in distress. It sounds like you did the right thing, just quickly bringing it to a more secluded, ideally shady spot (in case of something like heat exhaustion- in which case the bird may have its beak open/panting). While best avoided when possible, if the bird's in a dangerous spot like that handling it for as long as it takes to move it isn't inherently bad if you do so delicately. It's stressful, certainly, but the situation it's in is already stressful. While there's many things that could be wrong here, one possibility is
HPAI (highly pathogenic avian influenza), a deadly and highly contagious disease which all birds can catch. As such, make sure you wash your hands (and anything else that may have come in contact with the bird) off real well after events like this- the disease can be a death sentence for domestic poultry and other pet flocks. Also, without knowing your location, my best guess for species is wood thrush.
Determining the difference between Bockings 4 and 14 is done by consensus. It's like trying to identify the difference between twins.
"There are other spots on the web to get my fix proving someone is an idiot but no other place for what I get here." -- former permie Brice Moss, 2012.