Rules of the Road, rule 5, states "Every vessel shall at all times maintain a proper lookout by sight and hearing as well as by all available means appropriate in the prevailing circumstances and conditions so as to make a full appraisal of the situation and of the risk of collision."
Believe it or not, this rule does not require a look-out to be topside 24/7 making continuous visual sweeps....
Can you cite a source for that interpretation? The language you cited clearly states otherwise.
The worst case risk of collision is if this tanker is headed directly towards us and we are heading directly towards them.
In my mind, the worst case is that you missed the freighter in your horizon scan, when it was still hull down, especially during the daytime. Maybe you were out of sync on the wave crests, maybe there is distortion at the horizon from humidity, etc.
As for the closing time, you seem to be grossly underestimating the speed of modern tankers and freighters. Speeds in excess of 20 knots are common, and some are even moving over 30 knots. As are many naval vessels. For purposes of your calculations, you should double those closing speeds and halve the time.
I have watched fast ships come within sight, and then disappear over the horizon, in far less than an hour.