I'm no expert on magma flow or thermodynamics, but really? I'm not sure what kind of floodgates dwarves use (except that they don't impede anything when open), but here's what we know:
When closed, magma comes into contact with the floodgate itself, but not the mechanisms which open/close it.
When open, magma comes into contact with the mechanisms.
Now, your assertion is that the mechanisms alone have enough thermal mass that currently flowing magma will freeze around them in a way which produces sufficient obsidian that they will no longer be able to operate? (Keeping in mind that mechanisms which can exert enough force to open/close the floodgate can probably power through a small layer of obsidian as well.)
The magma doesn't just need time to cool, the heat needs somewhere to go. The doors and tunnel/channel walls are already heated to magma temperatures at the surfaces, so it's really just the mechanisms. Once they're up to temperature, there won't be any further cooling, and the mechanisms themselves are in motion (opening the floodgate) when this process first starts, so there's also that to impede crystal formation.
My intuition is that it should be possible to design these things so they work, and we can assume dwarven ingenuity. (Of course, I'd think you could design it so the mechanisms aren't ever exposed to the magma as well, but we can chalk that up to dwarven stupidity.)