perlop says:

The "=>" operator is a synonym for the comma, but forces any word to its left to be interpreted as a string (as of 5.001).

It's not a matter of precedence, since . is higher precendence than , or =>. It appears that the above doc really does mean "any word to its left" rather than "its left operand" as I would have expected.

I suspect it would be infeasible to change it to "its left operand". It may be possible to give a warning where => does force a word to be interpreted as a string but the word isn't the full left operand. I'd recommend filing a perlbug report on this.