I'm not sure why you're applying to quant positions if you want to be a trader; the roles and the skillsets are different, and neither directly "leads" to the other.
That said, depending on the bank, a lot of exotic traders are ex-quants or at least have a very quantitative background, similar to quants. This does of course not mean that most or even many exotic quants become traders or that the move is easy. Similarly, though I've seen people move from quant to trading in the etrading functions of banks, it's not an everyday occurrence.
The well-trodden career path of a quant at a bank (though as always the details depend on the firm, as do corporate titles) is associate for ~3 years -> VP, followed by promotion to ED/Director after a further ~5 years (though this is not a given in the same way a VP promotion is, and the timescales vary more). Once you make VP you are seen as somewhat senior/experienced, and this can make it more difficult to move to a different kind of a role (for example, I've seen some take a "demotion" back to associate just to move to trading, and even then things don't always work out long term - a good quant doesn't necessarily make a good trader), and so when people make big transitions, they typically do so early in their careers.
Moving to the buy side, into a quant role, is very common regardless of whether you're doing exotics or systematic market making, though the kind of shops you'd target are usually different, e.g. Citadel vs Citadel Securities. It's for this sort of a move that a name like GS may carry some weight on your CV, as it's easier to connect with former alumni of the same institution (especially true for smaller shops that were founded by e.g. ex-Goldmanites). When the work's similar, though the environment different, it's not that difficult to move even when you're a bit more senior (and slightly more senior people are even preferred, as there's not necessarily going to be much if any training on the job).
I've also seen quite a few quants move into structuring, model validation, and to tech both within and outside of finance (often with a data science/ML angle as that's the sexy thing nowadays).
Now the systematic market making role is going to be more tech-y as a job, and it's possible that the quant team is merely acting as tech support to the traders, thus gaining little understanding of the markets. It is also similarly possible that someone working in exotics gets stuck supporting old models they don't fully understand, or have the time to understand as their time is spent on patching rotting infra and responding to ad hoc queries. As a new joiner you're always going to be taking a bit of a punt, especially if you don't have the network to give you color.
I agree that if you go with the internet meme zeitgeist, systematic market making does sound sexier as everyone and their mother is doing it in a quantitative capacity. That said, markets are often about becoming highly specialized and finding a particular niche (and from this point of view the "brand name" of an investment bank - usually driven by IBD, not S&T, anyway - means very little), and so the choice between the roles, with the information given, is not entirely clear cut. Similarly for an outsider it is not obvious why there's been lots of
exits from Barclays' etrading function, and if this creates opportunities for new joiners and for those who stayed, or if it is e.g. ahead of looming cuts. Perhaps not a concern for someone who is already looking for exit opportunities before even joining (if IBD-like generic "exit opps" is indeed the main angle, maybe check via LinkedIn etc. where people have ended up from the offered roles).
With all that said, once you deviate from the "usual" career path of a quant, it's tough to make generalizations. I see a lot of the most interesting jobs being dealt through a good network and via luck to open-minded individuals who are not risk-averse.
