So, you'd prefer to avoid assignment,

I don't know what side effects you are talking about for assignments.

print statements,

I'm not fond of how print changes $!, but it's much more convenient than the alternative. There's also a very high level of consistency with other like functions. system is where it goes overboard ($?).

regular expressions,

I don't use $1, etc. I use the match operator in list context instead.

and even fetching the value of a variable?

Memoizing is a harmless side-effect.

I see as many side-effects as the code using 'for'

As I see it,
The purpose of for is to affect something (necessarily external) repeatedly.
The purpose of map is to transform a list.
You may see things differently, but it was clear that I was explaining how I see things.