Are readers supposed to root for morally ambiguous/decidedly immoral protagonists?
If you want to. Same thing applies to moral protagonists, antagonists, vehicles, inanimate objects, or creative wall ornaments. As a reader, you’re under no obligation to do anything. You don’t even have to keep reading if you don’t want to.
You root for the characters you choose because you want to. Not because of some arbitrary criteria was met. No one else can tell you who you should or shouldn’t root for. It doesn’t need to be the protagonist. Sometimes you’ll root for the villain. That’s cool.
Getting you to root for a character is entirely on the author. If you’re the writer, keeping your audience invested in your work is your job. Getting your audience to root for your characters is your job. You do that by making your characters compelling and interesting. You can’t make them like your, “cool” character just because you want them to, and audiences tend to be fairly resistant to overselling characters as, “so damn awesome,” in an attempt to sell them. Just look at the 80s and 90s comics industry if you want to see how badly this fails.
As the reader, you have no obligations. If someone tells you that liking a specific character or relationship is compulsory, they’re wrong. If someone tells you that endorsing some unsupported relationship is mandatory, they’ve disconnected from reality.
Root for the hero if you want. Root for the villain if you want. Hell, root for both. I’m not the boss of what you find compelling, and neither is anyone else.
If you think something’s trash, or unappealing, you can put that down and walk away. It doesn’t matter what some unhinged fan tells you. In the end, it’s your time, you don’t need to commit it to something that doesn’t interest you.
There are certainly reasons to read something you don’t like, or don’t enjoy. There are excellent works of art that don’t appeal, or are downright uncomfortable. Sometimes there is a purpose to perseverance. Just because you don’t like something doesn’t mean it’s without merit. With that in mind, sometimes it is worth sticking it out and finishing book. As writers, we’re strengthened by the things we dislike or disagree with as much as the things we love. In fact, reading things that challenge your views and perspectives can help you grow as a writer. However, aside from, “read something,” you have freedom to pick what you want. Just, for a balanced literary diet, pick and finish some things you might not like, read them, and try to form a comprehensive critique of them that extends beyond, “this sucks.”
When it comes to reading, do not let another dictate how you engage the material. Especially not someone who’s, “a fan.” They may have a better order to experience the material in, but ultimately how you engage with it, what you take away when you’re done, how you view it, is all up to you. Others may offer insights or opinions you appreciate or disagree with, but if you adopt them, again, that is your choice; no one else can tell you what to think about the story you read.
If the writer did their job, they’ll get you rooting for who they want you to. If you don’t like a character, you’re free to choose that, and no one else has any authority to tell you otherwise.