Why not write $\sqrt{3}2$?

The format $\sqrt{3}2$ is easliy confused with $\sqrt{32}$.

I also suspect that many early typesetters would skip the overline, so that $\sqrt{3}$ would be typeset as $\sqrt{\vphantom{3}}3$. In that case, $2\sqrt{\vphantom{3}}3$ is unambiguous but $\sqrt{\vphantom{3}}32$ highly ambiguous.

One possibility - would you rather think of the number as "two of the thing known as $\sqrt3$," or as "$\sqrt3$ many of the number two?"

Certainly one can find old books in which $\sqrt{x}$ was set as $\sqrt{\vphantom{x}}x$, and just as $32$ does not mean $3\cdot2$, so also $\sqrt{\vphantom{32}}32$ would not mean $\sqrt{3}\cdot 2$, but rather $\sqrt{32}$. An overline was once used where round brackets are used today, so that, where we now write $(a+b)^2$, people would write $\overline{a+b}^2$. Probably that's how the overline in $\sqrt{a+b}$ originated. Today, an incessant battle that will never end tries to call students' attention to the fact that $\sqrt{5}z$ is not the same as $\sqrt{5z}$ and $\sqrt{b^2-4ac}$ is not the same as $\sqrt{b^2-4}ac$, the latter being what one sees written by students.

It's simply a matter of clarity. If you write $\sqrt 3 2$ meaning $2 \times \sqrt 3$ rather than $\sqrt{32}$, it would be clearer to write $(\sqrt 3) 2$ or $\sqrt 3 \times 2$, but then you have to say: oh, what the heck, just go with $2 \sqrt 3$.

Another thing to consider is that neglecting to properly extend overlines is a tell-tale sign of a TeX novice. As you are already aware, to get $\sqrt{32}$ you need to write \sqrt{32} in your source.