I think you've mistaken with the different director positions (as well as their payscale) in PA.
DCD & CD are on the ground (Stationed in COs & CCs) and their salaries are as what you've mentioned ($5k-$6k for DCD, $7-$8k for CD) to manage a team of 15 to 20 staff. In comparison to the private sector, this makes them akin to an AM / Manager level I suppose.
Above the CD there is the GCD (Group Constituency Director), who bascially looks after all constituencies in the
GRC with the respective CDs reporting to him / her.
At the HQ level and above the GCD, there are the DDs (Deputy Director) and Directors for each of the 5 districts in Singapore (North-West, North-East, South-West, South-East & Central).
All GCDs report directly to the DDs and the Directors themselves report to even higher level Asst Directors / Directors above them (Grassroots Planning, Field Engagement, Partnership, Policy & Corporate development etc etc).
A district DD will already be in the payscale of minimum $10k. All the other higher level directors most certainly will be in the $11-$15k range as you've mentioned, If not, more.
That's the reason why the CMs, DCDs & CDs have the word "Constituency" in their titles, as they are just taking charge of their constituency. Shouldn't get their titles confused with the "Real deal" Managers and Directors in HQ level.
Hope this will be useful for people considering to apply with PA, that the DCD and CD position is not the end of your career once you've reach this position, cheers.