That wasn't actually always true. In many places there were no race requirements for suffrage and if you were a "free negro" who met the other then-qualifications (male, land owner, etc.) then you could vote. Of course, hardly any black men of the time did, not least because they didn't even have citizenship.
This is how we, unlike most other countries, ended up with birthright citizenship after the civil war. It was another compromise. The former slaves don't get citizenship but their children do.
And the issue with counting non-citizens is still true today -- the census still counts non-citizens and federal representation is apportioned based on the census, so the states with large non-citizen populations get representation disproportionate to their number of eligible voters.
Assuming the local official followed the law as written (hint: they didn't). Much like they would have for most of the post-Civil War, pre-Civil Rights Act era, they would have simply come up with an excuse not to let the rare free blacks who met the legal qualifications vote.
Since this is one of my common beefs...
The compromise was written at the behest of the Northern, non-slave states. The Southern slave-holding states would have been perfectly fine with five-fifths representation for each slave. Further, the point of the compromise was to limit the Congressional power of the Southern slave-holding states and was not intended as a value judgement as to the worth of slaves.