So that would rule out Humbert Humbert, Macbeth, Milton's Satan, Dorian Gray, Patrick Bateman – it's easy to go on and on. Do we really want a literary world where these guys don't exist?
Sympathy is one way of engaging a reader with a character, but it is only one way, and the most obvious way. I sometimes wonder if these commentators that insist on sympathy for protagonists actually mean empathy, which is a far more interesting reader response as it allows for a far greater range of characters that people can relate to - they can understand the character, without feeling compassion for them, which is what sympathy, a rather more patronising response, implies.
What is most important for a story is that characters achieve some kind of balance – the way I see it, a long running character that doesn't go through a character arc (this is typically the case with series detectives) need a balance of both admirable qualities and flaws – Poirot's vanity, Morse's moroseness etc. More traditional MCs that are bad, or unlikeable, need to attain some kind of balance, either by getting their comeuppance in the story, or realising their errors and changing. This is far more important than making a character 'sympathetic' – the need to provide a satisfying re-establishment of stasis.
It is lack of depth in a character, not lack of sympathy, that will cause readers to lose interest. The key to producing characters that readers engage with, or empathise with, is to drill down to their core humanity. We don't like to admit it, but we've all thought about killing someone. In the heart of us all, a bad man lies dormant.
It's also worth remember that in the world of a bad person, their actions are the right ones – they just play by different rules - if you lay those rules out, they are ones anyone can understand.