Tone is important. When you say things the right way, it can increase the number of people who are willing to listen to you.
But that only goes so far. No matter how good you are at framing things, some things that need to be said will upset people who feel entitled to be comfortable. And, when you upset people who feel entitled to comfort, they will lash out at you. This is not your fault; it is theirs. Tone has its limits.
Also, getting tone right is really hard. No one starts out good at tone; it’s a very difficult skill that you can only learn with practice. And the only way to get practice is to spend a lot of time talking to people about controversial things. Which means that, in order to get good at tone, you’re going to have to spend a lot of time talking about these things while you’re still bad at tone.
People who mean well and genuinely want you to be heard understand this, and will encourage you to keep speaking up and keep working on your skills at speaking up effectively. People who want you to shut up about the things you’re talking about will try to make you feel horrible about your tone and convince you that your tone means you have no right to say anything.
Sometimes, when people say that you should be more careful about tone so that you can be heard, what they really mean is “I don’t want to hear that, shut up and say something else I’m willing to listen to”.
Don’t believe those people, and don’t shut up. The most important thing is to keep talking. If you are bad at tone, some people will refuse to hear you. If you are good at tone, some people will still refuse to hear you. If you say nothing for fear of getting the tone wrong, no one will hear you.
Shutting up won’t get you heard. Speaking up might.
I am very bad at tone. I nearly lost my job because of not using the appropriate tone with both staff and students. I am sure some people don’t like me - and I think this is a major cause of my social anxiety - because of the tone I use for even non-controversial things. But I have a lot of feelings about controversial things that I avoid communicating because I know the tone I tend to use will upset people/make them feel uncomfortable. So, this is a really important skill that I need to learn, else start wearing a badge that says ‘the tone I use probably won’t be appropriate but please forgive it and listen to me anyway’.
If you’re having trouble with tone in professional contexts, I’d suggest reading through the Ask A Manager blog. She has a lot of really helpful posts on how to communicate in professional settings, including how to give and receive effective feedback.