It does when the outputs being spent are coming from different addresses. But sometimes (due to address reuse) all the transaction outputs are from the same address, like when people keep receiving payments in one address.
If you have reused addresses, you have already negatively affected your privacy.
Doesn't consolidating hurt privacy? Maybe consolidate, then mix? What's your opinion on this?
Yes, one "cost" of consolidating unspent outputs is a reduction in privacy. This should be considered when deciding if you want to consolidate your unspent outputs. The value of privacy is not infinite.
I don't think consolidating unspent outputs and subsequently mixing the resulting coin would provide much privacy benefit. When you consolidate unspent outputs from two previously unconnected addresses, an observer would know that the two addresses likely belong to the same entity. Mixing your consolidated coin does not change this fact. If you want to maintain your privacy, you will need to move your unspent outputs through a mixer before you consolidate your unspent outputs, or you could send your coin to multiple deposit addresses to a mixer, and have the mixer send you a single output. If you engage in the above, an observer may still be able to draw a link between the inputs and outputs in the mixing service.