Have you tried: "That was rude. Please say that more politely." 6 year olds are still working on emotion in language and sometimes need direct instruction to understand how their tone comes across. I remember being yelled at for being 'sassy' and being very frustrated that my parents weren't hearing what I was saying. If they'd taken the time to say "I understand you're upset, but your voice sounds very rude" and given me some suggestions for rephrasing, I think it would have helped. We do this a lot for our 6 year old and she is getting better.
For the 'or what?' sometimes I say the truth "Or I'm going to be really angry."
Have you read "How to Talk So Your Children Will Listen" by Faber & Mazlish? She's at a really good age for the ideas for this to work. I'd also recommend Kids, Parents & Power Struggles by Mary Sheedy Kurcinka. My all time favorite book is Playful Parenting because it talks about the need for connection before correction. I do find that my 6 year old is much more reasonable when we've had time for connection.
The final thing that I'd say is that I'm hearing different messages in your post, and I wonder if she's also getting mixed messages? You say "No consequences or punishments seem to really phase her. We can send her to bed early, put her in time out, take away privileges.." and then you say "I don't want to give her the impression that grown ups can force their will on smaller people simply because they are bigger..." So it sounds to me like sometimes you're doing punishments, but that you're not very comfortable with them. What's your underlying philosophy/message that you want to send? What's your ideal for how you would deal with this? I'd start there and work your way backward.
For example: I'm OK with timeouts in our house because sometimes we need separation and because I'm very sound sensitive. I can't stand to have dd screaming in my ear, so I send her to her room if she's being completely unreasonable. We do connect after, but in the throes of a tantrum, she's not ready for that connection. Dd is going to go to bed 15 minutes earlier than her brother today because she threw a major fit when it was time to get out of bed this morning. The 15 minutes earlier is because she clearly needs more sleep, not because of the fit. Dd was having a fit yesterday and kicking the wall (yes, the child really does need more sleep). I looked at her and said "If you kick a hole in the wall, you will need to help fix it." She stopped kicking the wall. If she's made a hole in the wall, she would have contributed to the cost of the repair from her money, spent some of her play time with me going to get the supplies and in helping patch/paint the hole.
We rarely use removal of privileges, and only for things that are directly related to the offense. So if ds and dd are arguing about who gets to play on the computer, the computer goes into timeout for an hour or so (i.e. we remove the privilege of using it).But I don't believe removing privileges for an unrelated offense teaches my child anything other than that random bad things happen when you act out.