The only way to avoid accidentally offending people is to just try your best to keep up with current discourse by having a diverse friend group and talking to people. Sometimes you might slip up by not knowing something is considered inappropriate now that used to be seen as okay, or being ignorant of something, like for example if you watch a lot of 90s sitcoms and English is your second language, you might innocently use the word "sp**z" without knowing it's now offensive. Or if you're not from the UK, you might say Scottish people are "English" rather than British and make them angry. In these cases, the important thing is to just apologise and change it if you can, or take down the offensive work or put up a content warning, NOT to dig in, get angry at people for being offended or to make a big post about how you're not changing it because you think people who got offended are wrong.
Because the thing to remember is, most people who everyone gets really angry at aren't being accidentally offensive any more. They've usually been told politely and refused to change it, or even doubled down.
And on the illuminati thing... er... who is offended by the concept of illuminati? Like... even if it DID look like the "illuminati symbol" (which it doesn't particularly, it's just an eye, kind of reminds me of the Inquisition symbol from Dragon Age), are members of the illuminati an oppressed group who are regularly struggling with being dehumanised or negative stereotypes that stop them from getting ahead in life? Er... no. Like, even if we assumed they existed, they'd be the most powerful people in the world, a tiny cabal of people with incredible political power and the idea that you're making their lives hard by.... putting a symbol vaguely similar to theirs on a character just doesn't make sense.
When thinking about what characteristics and symbols you should be careful of in character design, you should think more in terms of:
"Am I promoting negative stereotypes or dehumanising a marginalised group?" - ie. If all your Irish characters have bright red hair, comically exaggerated accents and are foolish alcoholics. OR there's only one autistic character in the story, and they're cold, robotic, emotionless, only care about their obsession with Star Trek and creep the other characters out, functioning only as a burden or problem.
"Am I promoting or glamorising real life symbols, phrases or words associated with hate groups?" ie. Do your good guys wear equipment that very clearly evokes German WWII uniforms and it's presented as really stylish, intimidating and sexy?
Try to consider whether your work could lead to people who often face violence, bullying, oppression or being kept out of opportunities to have a worse time by encouraging people to believe stereotypes about them or glamorising groups that hurt them rather than fretting over like.... "does this symbol look maybe a bit like a symbol some people decided is the symbol for a shadowy all-powerful organisation they believe exists?"
