You are getting a lot of disagreeing comments, but I do believe you're right. No, it isn't that big a deal to pick up a new alphabet, especially not a closely related one like anything in the Latin/Greek/Cyrillic family. But yes, it does make a difference in the absorbtion of a new language whether you are somehow submerged in it, and can run the osmosis routine on it.

Having had peek access to all Western European languages since the day I learned to read has clearly helped me gain early access to the style and idiosyncracy of each. Not that I can read them as such, but I can with some confidence pick out basic meaning from simple text in more or less all Indo-European languages written in some kind of Latin letters. For very basic meaning, this includes languages like Polish and Czech. It does not include Russian, unless I make some extra effort.

I actually know more Russian than Polish, but would no doubt have an easier time learning the latter. It does tend to matter what you had chiselled on your brain at age three or four.