Migrate ProcessingScheduleService consumers to enhanced interface

[Zelldon]

Description

blocked by #10001
blocked by #9999

After we have an actual ResultBuilder, which can be used without a LogStream dependency, we can use it in our classes where we need to schedule work and write records. Here we can replace the LogStreamWriter usage with the ResultBuilder. Before, we need the enhanced interface of the ProcessingScheduleService such that we can schedule Task which can return a Result.

Which classes need to be adjusted:

• Find classes that consume ProcessingScheduleService or StreamProcessor Actor and need the LogStreamWriters
• Adjust classes to use new enhanced interface
• tbd

[Zelldon]

Done via #10076