That's a good example! So the sqrt(n) space in the result must refer to space beyond that needed to write down the output--basically a working memory scratchpad. In which case, a string-copying function could use just a constant amount of scratch space.

I mean, for all I know, the result may have been proved in the model of decision problems where the output is always one bit. But I'd guess that it generalizes just fine to the case where you have to write longer outputs.

However, your question does make me wonder if the result still applies in the RAM model, where you can move to any spot on the tape in constant time. My theory knowledge is getting rusty...