It's interesting you mention vinegar and crushed coral. In essence, this is how a calcium reactor works. The reactor mixes CO2 with crushed coral and water. The result is a very low pH because of the CO2, which dissolves the crushed coral. The water then leaves the calcium reactor, now with the coral pieces dissolved into it, and the resultant calcium and carbonate gets added back to the aquarium via the reactor's effulent. To your point, some people conjecture that by dissolving actual coral skeletons, you're adding some sort trace elements or minerals that the corals may need, but I don't know if this has be conclusively proven. With the advent of ICP testing like that offered by Triton, it would be interesting to see if the reactor effulent actually does contain elevated trace mineral levels.
I suppose the same principle could be achieved with vinegar, however, I think the main issue is that vinegar is a very weak acid. Store-bought vinegar is typically only a 5% acetic acid solution. This would partially dissolve the outer layer of the crushed coral, but I don't think it would completely dissolve all of the calcium carbonate into solution before the acid was neutralized. The trouble then is that you're dosing a lot of an organic carbon source (vinegar), which may or may not be desirable in your situation.
If you really want the minerals that may be locked in the coral skeletons, I'd go for a calcium reactor. Keep in mind that while these devices work very well, they are very complex. Much more simple methods (2-part, kalk) exist for maintaining calcium and carbonate levels. Personally, I think a Ca reactor is a lot of hassle just for these ancillary "trace elements." Which again, I don't know if we're really sure if they actually are present in any meaningful quantities. But that's just my $0.02.
