I think there are strong arguments to be made for redistribution, especially in order to
enable equity, but I agree that redistribution and equity aren't the same thing.
I suspect they've become conflated in some circles because redistribution is seen as a real solution to lack of equity. Notably, this is literally baked into our system in terms of federal income tax, but many people argue that it doesn't go far enough. For example, redistribution of wealth could result in making childcare available to most people, eliminate childhood poverty, etc.
I don't think that in principal that 'redistribution' is controversial. Every contemporary society has some form of redistribution of wealth. However, many people see proposals around it as extreme. Though, whether the proposals are extreme may depend on one's knowledge of the consequences of action (and inaction), among other things.
This gets sticky, because a lot of social activists get hung up on this to the detriment of making real-progress on things that are not controversial when presented in the right way. I see a lot of calls for "ideological purity" and "absolute dedication to the cause". These calls, among activists, often become central to the conversation, as opposed to specific policy proposals that could make a real difference. Personally, I find it rather upsetting to see this happen, as I really do see the problems that are caused by lack of equity, structural racism, etc. But the way the dominant activist groups push for change seems to be too focused on ideology, as opposed to pragramatic solutions. I see the value in anger, which turns into dedication and motivation, I think that it sometimes goes way too far, and then becomes a form of unfortunate self-sabatoge. Anyways, I think I've digressed a bit.