if you're a Democrat, you probably don't see the use of school shooting incidents as justification for laws that violate the Second Amendment to be examples of "for the children", while I do.
That depends on what you mean. If you're talking about the sudden push for banning guns that happens after every school shooting, then yes I do find it to be one. If you're talking about using the relative statistics of school shootings in the US vs. other countries, then I find that to be a valid, and not a "for the children" argument, as it's typically brought up with other statistics to cover the entirety of gun-related crime/etc.
As an aside, you're stretching the Second Amendment's meaning if you think that all gun control violates it; not even the NRA thinks that. I live in a state where it's easier to get a gun than a car, and I find that to be ridiculous. I would be for tighter licensing controls, at least to the level of cars - why do I have to prove I can safely drive a car before I'm allowed a license to drive, but I can wander around in public with a gun without any kind of license, or that I can buy one without any evidence that I'm even remotely competent at gun safety?
Likewise, the "have the government make this decision because it will decide more intelligently than the parents can" argument is almost-entirely a Democrat thing. It's not a "for your children" argument, it's a "for those other parents' children
Once again, that's a false argument. Nearly everything the government does - in fact, pretty much the entire point of having a government involved in any kind of regulation - is to do this. The point is minimum standards, and the general safety of the citizens. And the fallacy you're bringing up here is that what parents do with their children only affect their children. The current anti-vaxxer bullshit is a perfect example of this - the government didn't step in when it should have (by requiring kids going to school to be properly vaccinated), and now we've got measles and whooping cough and other nearly-eradicated diseases having major outbreaks. These outbreaks put more than just those unvaccinated individuals at risk - they also affect those with weak immune systems, babies too young for vaccines, people who got the vaccine but it just didn't take, etc. by interfering with the herd immunity.
All sorts of health, safety, and education mandates fall into that category, such as the Democrat hostility to alternatives to public schools.
Hate to tell you this, but that hostility you're talking about isn't just Democrats, and among Democrats it isn't even close to a universal belief. I am assuming you're talking about diverting public funds to private schools here, in which case the typical reaction against it - which I've heard from Democrats and Republicans alike - is "if the school's not doing good enough because it's underfunded, why are we taking money away from them instead of using it to fix them". Unless you're talking about the whole "public schools are actually liberal brainwashing programs made to teach kids that Jesus isn't real" thing I hear occasionally - at which point you're a crazy fucking idiot and have no idea what you're talking about.