In point of fact I have flown on SEARCH MISSIONS looking for ships on the open sea with the aid of surface search radar to boot and it is you who is incorrect claiming a high POD for objects 30 miles either side of one's trackline. That is feasible but in the real world is wishful thinking.
OK. Feel free to disagree. I admit AMERICANS won't spot very often in THIS age (WWII is different).
When Elat was sunk by Styx missiles (in 1967) her lookouts spotted them inbound - the enemy never even left port to attack the destroyer at sea.
I was assigned to solve the problem of detecting and engaging, which was hard, becuse our search radar could not see missiles (they are too small) and we had no confidence American observers would either. But CNO at the time believed (correctly as it turned out) that we could detect the missiles radars - they needed to find a moving target - and often the radars of the unit which would fire the missiles - IF we learned to "think passively" (which is almost as un-American as being a diligent observer).
Even so, I trained special observers to WWII era Japanese standards, although we didn't have as good optics, we nearly did. I feared missiles guided by infra red (they were in fact made but not given to North Viet Nam - something I didn't want to depend on) - so I wanted a backup system. I have a British attitude (master the basics) - so we fielded even .50 cal MGs in the AAA role (found to still matter as late as 1982) - and regular mortars to help with distributing chaff (before chaff rockets were yet invented).
I will rephrase: IF I trained the observer, and the weather is good, and he failed to detect any surface vessel bigger than a small craft, I would consider shooting him. I expect him to spot even an airplane without fail at 50 miles - never mind a ship.
There is, however, something you were not taught (which I got from Saburo Sakai) - a visual trick called "focus on infinity." Frankly I didn't believe in it - but as a young sailor I visited an observatory (at Meadowbrook Hall) where an astronomer was teaching elementary school children. I asked him about it - and he taught us all - even children - how it works - so we could see STARS in daylight. If you cannot see a first magnitude star EVERY TIME there is one over the horizon in clear conditions, you will indeed not spot ships or planes at 30, 50 or even 100 miles. But people I train can and will.