The term is historical, not functional.
Before option-pricing theory, securities pricing was an art. Yes, there were capital asset pricing theories for fundamental analysts. But at banks, a phone and hustle were the tools of the trade.
With Black-Scholes (and put-call parity) came the ability to (a) manufacture options out of other securities and (b) print objectively-wrong quotes. The former gave banks an incentive to build the business. The latter gave them the incentive to automate. The people they hired to do that were quants.
The first generations of quants automated option-pricing models. Through the 80s, they found themselves involved in more products, e.g. securitised loans and mortgages. By the 90s, they were launching funds. Today, almost everyone on a modern trading desk is a quant to some degree.
Is there any answer to it yet ?
Not really. Black-Scholes (and its variants) tend to assume log-normal rates distributions. But that’s just a default, and one chosen with the explicit assumption of positive rates.
Most practitioners have had custom curves for decades; using one that pierces zero is a trivial modification.
You'd be more likely to find deriv quants at a bank selling such things to people, whereas the strategy quants would be what you'd find in the funds.
But even within that fund side I'd imagine the job looks very different depending on what firm you're at and what they ask you to do. Not everything is glamourous blue-sky "what should I buy or sell" stuff. A fair bit of it is things like cleaning the data. And there's going to be a lot of tooling. Since nobody says what they are doing, everyone has their own version of a research pipeline, and that needs to be kept clean. Likewise on the execution side, there's a load of code to be looked at, and a lot of data coming out relating to trading costs and such.