No. Sn+Sm would be n+m+2, while S(n+m) is n+m+1.

When I read about Peano axioms I mentally think of Sn as the function S(n) = n+1 and that helps me understand them better.

As a non mathematician sometimes I wonder if this substitution is causing me to misunderstand something else about them.

S is not a multiplier; it's a call of the successor function. The distributive property doesn't apply.