Cry of semi-despair: "I am stuck in a director-level position and can't get promoted"

The promotion to MD is very political. You move from a role where you have to produce/deliver, to a role where you manage: making sure people below you are working in the direction set by the clients (business, trading, etc..). Usually there is only on MD per team (sometimes co-heads but this is unstable) so to go up, you either need to evict your boss or he is also moving up (if he is good that could be happen but most of the time they have also hit the ceiling themselves). Your boss will never promote you to the same level then themselves (obviously they want to stay your boss). So you need a sponsor on MD or above, who behind closed doors will defend your case and show the rest of the world how things would be better off if you were to step up to MD. This is way every promotion are harder to obtain and the reality is that a lot of people just stay where they are for a very long time ... The rule is: you'll be promoted till you reach your level of incompetency...
Change job only if someone above you will always undermine you when your name is on the promotion list. If you change job, you have to start from zero again, prove your worth to the rainmakers above you...

Quant Dev is a broad title.

Are you involved in the strategies / algos, or just a dev?

If a developer - then could try and maybe push towards that side a little more

If already in such a role - what's the desk
Are you involved in alpha generating strategies
Or in risk management / modelling

If the latter I'd say that is not idea and more difficult.
Do think to better answer would need to know the above

One thing you could do is look for contracting work. I don't know what you currently make, I'm sure it's a very good salary. But you could go into contracting for a while, build up some management skills and earn a bit more money (and pay much less tax). Then after a few years you could go back into a full time role at a higher level, perhaps MD.

What they like to see in MD's is not only good skills for the job, which I'm sure you have, but also someone who is dynamic and has a little project management experience. That's why we see a lot of ex management consultants in MD-type roles.

That's just my thoughts.