This answer writes
You describe an imaginary measurement which could not be performed at the time of Columbus or before. There are several reasons for this. 1. Atmospheric refraction at the horizon is about 30' but it is not constant (depends of the weather conditions), and the knowledge of refraction at that time was very rudimentary. It is discussed by Ptolemy, but he could not really measure it, and I am not aware of any reliable measurement until 17 century. 2. One needs a devise to measure small angles with sufficient accuracy. We are talking of angles of less than 1 degree. There is no evidence that such devises existed until 17 century. (If you read how Archimedes tried to measure the diameter of the Sun directly, you get the idea of the difficulties involved. There was certainly no progress in this since Archimedes to the time of Columbus). 3. "You sail until you see 1/2 of the mountain". How do you measure what distance you sailed? Read the book of Morrison about how they measured sailing distances at the time of Columbus, and see how crude the methods were.
So it incorrectly claims that the atmospheric refraction at the horizon of 30' applies to mountains (rather than just celestial bodies, for which this number is calculated). The refraction is actually caused by differences in air density, so light coming in from outer space will experience very different total refraction than light coming from a 1000m mountain.
Another clear mistake in this answer was claiming that the simple method I described in the question required explicitly measuring the distance one has sailed. That is incorrect. I even gave a formula: $$ h \over {2 \alpha ^2} $$ It does not need this distance measurement.
I pointed out these mistakes in the comments, but they were deleted by someone. Who deleted these comments and why?
There were other mistakes in the answer BTW, but I guess I won't bother, if this SE would rather lap up nonsense.