The constitution doesn't say one has to be a citizen to vote either.
A very strict interpretation of the 14th amendment would re-write a number of laws and case law on what non-citizens would be allowed to do, especially 1st amendment rights (off the top of my head)
Only someone who doesn't understand the US Constitution, Article I, Section 3, Clause 1, Article I, Section 4, Clause 1, the 15th Amendment, the 19th Amendment, the 24th Amendment, and the 26th Amendment.
In a nutshell, the federal government doesn't hold elections that involve the general population. That duty falls to the states. Only after the polls close and the votes tallied do the states actually vote in the electoral college. The Senate was an exception, where the state legislatures picked their members to be in the US Senate. The 17th Amendment changed that to a popular vote at the state level to match the House elections.
So, in a nutshell, the Constitution doesn't have to specify only citizens may vote, because the Constitution governs the feds. And since the Feds don't hold elections outside of the electoral college, your point is moot.