It's all in the Certification
The reason for Boeing having you manually deploy the SB is simple:
The accelerate-stop distance published in the AFM, and on which all takeoff calculations are based, is based on Maximum Manual Braking and Speedbrakes deployed. The reversers are a bonus not accounted for during calculations.
Therefore, Boeing wants you to first do the two things that will absolutely guarantee that you stop on the runway: Brake and get the SB out.
In order to safeguard for pilot error, they put in RTO which will brake for you if you forget to, and auto-SB which will get them out for you if you forget them.
The auto-SB is intended as a safeguard and not as a primary means of deploying the SB.
Changing the QRH and the order of critical action items during a rejected takeoff is possible, but the airline has to take responsibility and have it approved by their CAA.
For a system to be considered as reliable enough to serve as primary means of operating something, and basing critical takeoff computations on the assumption that it will operate, it has to be of a certain reliability level.
I do not know if the auto-SB system meets these reliability standards to be considered as a primary system with the checklist only having you verify it's operation.
Do you?