The condition in the linked thread is also necessary: if $(\mu_n)$ is tight, then there exists an increasing sequence $(R_j)$ with $R_j\uparrow+\infty$ such that $\sup_n\mu_n(\mathbb R\setminus [-R_j,R_j])\leqslant j^{-3}$. Define for $x \in [R_j,R_{j+1})\cup (-R_{j+1},-R_j]$, $f(x):=j$. Then $$\int f(x)\mathrm d\mu_n\leqslant \mu[-R_1,R_1]+\sum_{j\geqslant 1}j\cdot \mu_n\left(\left[R_j,R_{j+1}\right)\cup \left(-R_{j+1},R_j\right]\right)<+\infty.$$ However, I am not sure that this is simpler to check than the usual definition.