I'm a bit confused at the moment, but is it possible to automatically adjust start times of all tasks corresponding to the start time of the parent project?
Also, when adjusting the project start time later the task start times should follow.
I would like to achieve this without using Fusion.
Are you asking specifically about the start time, not necessarily the start date? I've found that the first task generally aligns with the start time of the project, but then depending on constraints and durations of remaining tasks the start time of subsequent tasks can push out from the time that the first one starts especially if you have any durations that aren't full days.