I think a villain doesn't always need a sympathetic motivation, but they do generally need a justification they tell themselves that vindicates their actions from their perspective that if the audience hears it, doesn't fall flat or fall apart. Some people will justify literally anything to themselves with excuses like, "I'm just trying to survive in a world full of bad people," "I'm just doing what's best for people even if they don't see it," "I was wronged and am serving up justice." etc.
An example of a villain whose actions don't make sense with his motivation is the antagonist of the first Iron Man movie. He starts off very promising. He's Tony Stark's second in command, he likes making money and having the shareholders happy and he's not happy that Tony has randomly decided after a brush with death, to become less weapons-focused and more philanthropic, making the company potentially less profitable, threatening the lifestyle of him, his shareholders and maybe even employee jobs. Okay, great! Motivation makes sense, it's probably only sympathetic if you share his hyper-capitalist mindset where making weapons to kill people so you can make more money than anyone ever really needs is somehow good... but you know, there are people who think that (like people who think the economy is so important it's worth letting loads of people die from a lung-destroying virus........). The problem is, his actions: messily trying to assassinate Stark by building a giant mech suit and going on a rampage... make no sense with that motivation. It actively undermines his goals. There's no way people are not going to violently take down the dude who went on a very visible, public mech suit rampage and put him in jail even if he does kill Tony and then it'll be bad for him and the company. They just wanted a big cool fight at the end of the movie and contrived a shaky reason.
Meanwhile, Thanos is generally agreed to be a good villain. He wants to save the universe from the misery of overpopulation and limited resources, and he wants to do it in what he sees as the fairest, most humane way. A lot of people who watch the movies even think he's right and that the heroes are reacting in a very emotional way to a logical problem. To himself, Thanos is a tragic hero, the only one willing to shoulder an enormous sin for the good of everyone, but to his daughter, Thanos is a controlling and abusive egotist, willing to sacrifice everything but himself for a grand destiny he has decided on without consulting for other options because he holds other people in such low esteem.This ultimately egotistical motivation wrapped up in noble sounding goals, is laid bare in the end when he decides to just wipe out everyone to create a "grateful universe"; it was always about control, so even if you think it was the right thing to do, when you really get into his head, his motivations weren't completely sympathetic. He didn't just want to save the universe, he wanted to BE the saviour of the universe and to do it exactly his way and for people to be grateful.
In the case of Urien in my comic, Errant, he thinks his motivation is sympathetic; life is SO unfair, he was born after an age of knights ended to doesn't get to have cool magic powers like his parents, and has to be just some nobody even though he's so handsome and such a talented knight and he's saddled these two squires, one of whom has no chance of going anywhere, and the other constantly questions his judgement! AND his parents are dead and he's under the care of his weird old gay borderline alcoholic aunt! Boo-hoo-hoo! Doesn't he deserve a chance to show everyone how great he is!? Shouldn't teenage girls know their place instead of shooting him down all the time and being all morally righteous!?
His narcissistic attitude was very much based on people I've observed in life (and some... politicians....) or had to deal with in a professional context, and I played into people's base assumptions early on; that a handsome, charming man who speaks with clarity and confidence must know what he's doing and ultimately be heroic and justified, even if he's an opinionated jerk who's not actually that profound and is really quite condescending to girls and women, and also the inbuilt assumption that teenage girls are annoying and their opinions are inherently half-baked and naive. There were even people in the comments agreeing with him for giving the girls a talking to, and even now he's a full-on villain, readers do admire his charisma and confidence. He's always got an excuse, "We have to protect our land by being strong", "this is part of our culture", "I'm giving you a job", he will say whatever he needs to, but at heart his motivation is as simple as "I'm special and I deserve to have the world revolve around me and to do whatever I want because I'm smarter, braver, better looking and stronger than other people and they should see that."
Urien's feelings are pretty banal and universal, he just uses them as an excuse to himself for acting like an absolute dick. Most people can relate to feeling like they deserve more. Most people can relate to feeling affronted as some naive kid whose life is free from responsibility for the outcome of hard choices moralises at us or shows us up. The difference between the reader and Urien (hopefully), is that you have the humility and empathy not to decide that means you should just take everything and try to humiliate and undermine anyone in your way.
Basically villains are villains because even if they say they're doing things for a very noble or sympathetic or relatable reason, their selfish or sadistic nature will come through in what they do to achieve those goals in a way that makes their reasons ring a little hollow. They tend to make decisions on others behalf, dehumanise certain people or groups as "acceptable casualties", take their revenge or plans a step too far etc. Brooklyn 99 said it best: "Cool motive. Still murder".
