Yes, Democrats are the problem. Republicans have historically been supporters of public school, and continue to voice their unwavering commitment to public education and they are most certainly not painting them as liberal brainwashing institutions who "groom" children into becoming sexual deviants.
I get what you're saying when it comes to school funding - but you're conflating an issue of federal dollars being used for foreign affairs and a public institution who's funding comes primarily from the state level. It's borderline impossible to do even at times when education isn't attacked in politics, it's impossible to do now.
Injecting more federal dollars would require a massive federal bill similar to No Child Left Behind. Conservative voters - already convinced the Gubmint wants to brainwash their kids - would absolutely not support it. Individual states would likely push back on the federal government trying to change their curriculum or they'd outright reject the money just like they did with healthcare. Lastly, they'd need 60 votes to do it so it's simply not going to happen.