You are right that it would be inconvenient to wait for a particular day, but that is not what was in use as of 1805.
The
book I linked from 1805 includes the following methods for determining latitude, none of which require you to wait for a particular day, although some require you to wait for a particular time, and others require you to wait for some time interval to elapse:
- To find the latitude of a ship at sea, by the observed meridional altitude of a star (problem V)
- To find the latitude of a ship at sea, by the observed meridional altitude of the moon's limb (problem VI)
- To find the latitude of a ship at sea, by several altitudes of the sun observed near noon (problem VII)
- To find the latitude of a ship at sea, by two observed altitudes of the sun, and the interval of time elapsed between the observations. (Problem VIII)
All of these are in the Problems and Examples section near the very end.
So there is no need to wait for a particular day, or to rely on a single star - if the north star is obscured by clouds, any other star for which you know its location in the sky will do. There's no need to wait for night, you can observe the sun multiple times.
In that book there is a long list of place names and their latitudes and longitudes. The Royal Navy navigated to many of these locations using these techniques.
I would say that qualifies as having tested AND used as the primary method of determining latitude, wouldn't you?