An old result of Iwasawa is that in any connected Lie group $G$, every compact subgroup is contained in a maximal compact subgroup, and all maximal compact subgroups are conjugate.

Let $G$ be a connected Lie group and let $Z$ be a discrete central subgroup. Then $Z$ has an infinite torsion quotient $Z'=Z/B$ (lemma below). Write $H=G/B$. Then $Z'$ is a discrete central subgroup of $G$ and is infinite torsion. Let $K$ be a maximal compact subgroup of $H$. Every finite $F$ subgroup of $Z'$ is contained in a maximal compact subgroup $K'_F$ of $H$; since $F$ is central and $K$ is conjugate to $K'_F$, we deduce that $F\subset K$. Hence $Z'\subset K$, a contradiction.

This would be fine with a proof of Iwasawa's result; the only problem is that I don't know if it relies on the result of finite generation of discrete central subgroups!

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Lemma: if $Z$ is an infinitely generated abelian group, then $Z$ has an infinite torsion quotient.

Proof: let $B$ be a maximal free subgroup. Then $Z/B$ is torsion. If $B$ is finitely generated then $Z/B$ is infinite. Otherwise, choose an infinite subset $(e_n)_{n\ge 1}$ in $B$ and replace $e_n$ by $ne_n$ for all $n$, to obtain a smaller free subgroup $B'$, with $B/B'$ infinite torsion; then $Z/B'$ is infinite torsion.

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Actually, instead of all the power of Iwasawa's result, it's enough to prove the result using the weaker result: for any connected Lie group $G$ 
and any increasing sequence $(K_n)$ of compact subgroups of $G$, we have $\overline{\bigcup K_n}$ compact.

Possibly this can done by hand (say, without using such things as Levi factors, just more basic Lie theory), I'll think twice.