I don't think everyone is an idiot. It's a big assumption to say so. People have their reasons and motivations. Many people weren't given a fair chance in life, many lacked an education, many were raised to think a certain way or in a certain culture.
I'd wager I have tried arguing with the people I would categorise as 'anti-intellectual' more than 99% of people to ever have lived trying to understand them, and I did develop a level of empathy and understanding. But still it remains that just because people have reasons, they aren't necessarily valid and eventually people are responsible for their own self-reflection and decisions. 'Solving' this issue with people who already have engrained beliefs is incredibly difficult because they need to be responsible themselves. It isn't something I can actualise solely. It's far easier to start from a fresh generation, because changing is hard when you are used to something for so long. You see this in religious people especially and in people who eat meat. I know why they are/do what they do, but still it doesn't give justification. Many of them may even doubt their beliefs, but still cling to them. I know they do because I used to as well. I even still proceeded to do things I know were wrong. I don't claim to be flawless. Furthermore, I also know there is no reason to come in blaming these people ruthlessly because it will not progress anywhere and serve no purpose, what is done is done. But I cannot deny how annoying it is to see people still refuse to even attempt to learn.
The 'solution' is very complex if you want people to change because it will be an incredibly difficult task and something that would require an entire cultural shift to how people think. No doubt long term and I don't have the answer to how, and even if somebody did, it still relies on others to make a decision themselves. You can only make your own judgement of individuals for so long, soon enough you can recognise patterns and arguing every case is not possible with what time you have. I do my best.
I'll never not see anti-intellectual people as stupid, even if they have their reasons. I used to be an idiot who actively did things they thought were wrong. But eventually I stopped because I realised it's completely hypocritical and morally and logically wrong. I came to that conclusion without need of others judging me through my own self-reflection, and I'll admit it was hard. Even so, I wished somebody would have called me out, but I guess animal consumption is so engrained in society people don't even question it. I had my reasons to do so, but they were by no means a justification. I still try to understand things in different ways, but eventually it becomes redundant taking each case and doing so. The reality is that anti-intellectualism is incredibly prevalent and people need to change their ways of thinking. Sometimes they are just blatantly wrong and need to stop letting their emotions do their thinking. Sometimes there is nothing to understand. I don't know why people are so bent on seeing every individual separately, it's impossible to do so. Even if we do, they are still liable for their actions. Such as choosing not to self-reflect.