So who, in (contemporary, analytical) philosophy talks about true essences?

How is that relevant?

But that's inefficient. It's wasted effort to quantify what doesn't work conceptually. It may be impossible to always get the conceptual stage right first time, but one can adopt a policy of getting it as firm as possible...

Writing your intuitions up in a formal, precise way can often help you better understand what they are, and whether they're coherent. It's a good way to inspire new ideas and spot counter-intuitive relationships between old ones, and it's also a good way to do a sanity check on an entire framework. So I don't think steering clear of math and logic notation is a particularly good way to enhance the quality of philosophical thought; I think it's frequently more efficient to quickly test your ideas' coherence and univocality.