Word or expression for when a person not under pressure is able to see an issue easier than the person under pressure?

There is the phrase: a spectator sees more of the game

Trying to find the origin of that is proving particularly difficult.

The Chinese have a similar phrase: The person on the spot is baffled, the onlooker sees clear. I expect it's a lot more snappy in Mandarin.

The meanings of both are quite clear I think and, I think, apt.

The person under pressure is sometimes said not to be able to see the wood/forest for the trees.

In other words they get so bogged down with the details of the problem that they cannot grasp the overall picture. Sometimes a fresh mind quickly gets round the matters in hand through a change in perspective.

I have an example which is related to but not exactly what you are requesting.

Cooler heads prevail.

People who are calm and focused are the most likely to succeed or to have the greatest influence on events.

wiktionary.org

the ideas or influence of less emotional people prevail. (Used of a tense situation.)

thefreedictionary.com

This phrase suggests that those that are more impervious to pressure will make shrewder decisions, which is not far off your question.

Some single word(s) since the others are phrases.

detachment

The state of being objective or aloof: he felt a sense of detachment from what was going on

I say "word(s)" since the definition provides the synonyms objectivity and aloofness, though detachment seems to be the best fit for a situation where one is under the gun. Aloof implies being uncaring rather than involved. Objective implies logic rather than emotional entanglement. Detached best means not being caught up in the moment.

Someone explained this to me once using the analogy of tunnel vision. When your head is in a problem, it is harder to see the bigger picture. Someone appearing and looking over your shoulder sees the answer immediately as they aren't miles down the rabbit hole. You feel stupid, but you shouldn't because it would be the same if you had switched places.

I think that this is why when you 'come back to something' later, you see the answer, i.e. because you have taken the blinkers off, and you are seeing from a higher perspective. You have ascended the mountain and are looking into the valley, though from the valley you can also see things that those on the mountain cannot see.