Your conjecture is false. Indeed, let $P(X_n=n)=1/n=1-P(X_n=0)$. Then for all $n$ we have $EX_n=1$, $Y_n=X_n$, $Var\, Y_n=Var\,X_n=n-1$. So, $\sum_n Var\,Y_n/n^2=\infty$.

Welcome to MathOverflow! However, your conjecture is false. Indeed, let $P(X_n=n)=1/n=1-P(X_n=0)$. Then for all $n$ we have $EX_n=1$, $Y_n=X_n$, $Var\, Y_n=Var\,X_n=n-1$. So, $\sum_n Var\,Y_n/n^2=\infty$.

Additional note: Your statement that $EM<\infty$ does not follow from the Beppo Levi theorem, and it is actually false in general. Indeed, in the above example, by the second Borel–Cantelli lemma, the events $\{X_n=n\}$ occur almost surely (a.s.) infinitely often, and hence $M=\infty$ a.s.