You pretty much have it.
The new fan doesn't have a run-on timer, so if you want that feature, I think you need a different fan.
Yes, I think you are correct that the PCB markings show where a Switched Live would go, but the other missing components might also be required, so I don't think it's as simple as making a connection to that PCB pad. ( It looks like R6, D3, C2 form an RC timer, which probably drives the base of transistor TR1 to switch it off after the time has elapsed. It looks like TR1 has been shorted out C-E so the fan will always run. )
If you are happy with the fan just going on/off with no run-on, you need to connect the fan L to the Switched L, and that might be either the Black or Brown. Needs testing.
The fan N should go to the incoming Grey- Sleeved-with-Blue (9-o'clock). The Blue sleeve indicates the grey wire is not a phase, but has been re-purposed as Neutral.