Say we divide 1858.54 cm by 5 -- where 5 is an exact number. We get the quotient 371.708
According to the rules of significant figures, we would answer write 371.708 cm, since "1858.54" is 6 significant figures, and 5, being an exact number doesn't count.
My question is, how do you make sense of 371.708 cm, which is more precise than the original? That is, the quotient is precise to the nearest thousandth while the original is precise to nearest hundredth. To me that doesn't make sense.
Significant figures are a heuristic for keeping track of accuracy in a calculation. If you want to be careful, you need to assign an uncertainty to each number and keep track of how they build up. You also need to decide whether you want the error band on your result to be a worst case or some sort of "$3 \sigma$" based on assumptions of the distribution of errors in your input. Often we assume normal errors because they are easy to compute with, but real world errors tend to have longer tails than that.
For your specific case, absent other information we assume $1858.54$ is $\pm 0.005$. If we divide by an exact $5$ we get $371.708 \pm 0.001$. This is not quite as good as a $6$ significant figure $371.708$, which should be $\pm 0.0005$ but it is better than $371.71\pm 0.005$. For many purposes we do not care to this level of detail and we just track significant figures. If we do care, give the result as $371.708\pm 0.001$ and indicate whether this is a hard limit (like if the original $1858.54$ was measured with a graduated scale) or has some chance of being outside the range.
According to you, significant figures don't matter. What matters is whether you have hundredths or thousandths. (Or, presumably, tenths or hundredths, etc.)
So let's apply your logic in the other direction.
Let's try a simpler example. Take a measurement of $102.3$ and divide it by the exact number $100.$
If $102.3$ is accurate to the nearest tenth, the actual value $x$ that was measured is somewhere in the interval from $102.25$ to $102.35.$ Anything less than $102.25$ would be nearer to $102.2,$ and anything greater than $102.35$ would be nearer to $102.4.$
When we divide a number $x$ that is no less than $102.25$ by exactly $100$, the result is no less than $1.0225.$ When we divide a number $x$ that is no greater than $102.35$ by exactly $100$, the result is no greater than $1.0235.$
So whatever the true exact value of $x$ is, when we divide by exactly $100$ we get a number that is at least $1.0225$ and at most $1.0235.$
A good way to represent the resulting number without throwing away information is to write it as $1.023,$ accurate to the nearest thousandth.
If you write the result as $1.0$ (to the nearest tenth) then you have thrown away information.
For an even more extreme example: The number $12.3,$ accurate to the nearest tenth, divided by exactly $1000.$
This works in the other direction too. If you start with a number accurate to the nearest thousandth and multiply by exactly $100,$ you get a number accurate to the nearest tenth, not the nearest thousandth. Any error in the original measurement is magnified by the multiplication.
