The space defined by "# of heads in three tosses" is composed entirely of "not-numbers". We have to have some way of converting that over right?
But then you say "Ok well how hard is it to work with that sample space". And not hard at all. But what happens when you are dealing with six different outcomes counted over 1000 individual trials? Or what happens when your random variable is defined as something like time (measured discretely or continuous)?
I think the most significant use of the random variable is that it allows us to visualize the distribution of such items in a meaningful way that charts and lists of sample spaces and events couldn't possibly accomplish.