As long as it's possible, the insane ones might do it. We cannot allow an asteroid gap!

I imagine it would be possible to construct the system to substantially reduce the risk, even reduce it to "cosmic rays cause bit-flips which launch nukes" levels of risk. It's not like military nuclear command-and-control, you don't need one person who can push the button at any time. If we spot an asteroid, we'll have some time to deal with it. The system would presumably need to be constructed, which would take a few days even if everything was in storage on standby. Then, to activate it, you could have something like the US nuclear weapons' PAL, but instead of the codes being in the hands of one person, you spread them out. Just before launch it's activated with something analogous to the ICANN key ceremony: a bunch of well-regarded astronomers, rocket scientists, and aerospace engineers use their section of the Earth Asteroid Defence Key, and they press the metaphorical button together.

A system like this would require so many people at each step that I don't think one insane actor could do something like this even if they were the president.

It's not obvious that such a system would be useful for that at all. The effects on the trajectory would be hard to predict. It would be useful at all because any meaningful change in trajectory hundreds of millions of miles away from impact results in a miss.

Also such systems and missions are the results of thousands of people working together not the pres pulling up a web interface and clippy asking "So it looks like you'd like to deflect an asteroid today!" Much like jump starting the apocalypse with global nuclear war can't be effected by pressing a red button somewhere.