When writing a number such as $65$ in scientific notation, is it ok to leave off the exponent on the ten? I.e. $6.5 \times 10$?
In the slightly unusual scenario where you want to write this number, I think it should be $6.5\times10^1$. Leaving off the $^1$ would cause me (and probably most others used to seeing scientific notation) to parse the expression as a mathematical expression rather than a number. If you're unhappy with that, just write $65$.
It's not that unreasonable to sometimes write a number like $65$ in scientific notation. For instance, if I were writing a table with numbers with a large dynamic range in a column, I might write them all in scientific notation to preserve a nice alignment. I'd even write $1.2\times 10^0$, if relevant. I've found that since I got a good bit of practice reading numbers in this format, I don't even notice any more and my brain simply acquires the number without effort.
The use of scientific notation makes writing very large and very small numbers much easier.
However scientific notation is also very useful when one is concerned with the accuracy of a numerical value and in particular to show which zeros are significant.
$200$ can be written in scientific notation as $2\times 10^2$ or $2.0\times 10^2$ or $2.00\times 10^2$ to give an indication of the accuracy to which the numerical value is quoted.
What about your value of $65$?
That is different in meaning from a value quoted as $65.0$ and $65$ by itself will clearly indicate that it is the $5$ which the least significant digit.
So using $\times 10^1$ is not necessary for $65$ however $60$ is a different matter if you really want to show whether or not the zero is significant.
It might be that with $60$ as part of a table with other values present there is no need to use scientific notation but to be sure you need to use $6.0\times 10^1$.