I do not understand how is this invalid:

1, Timeline is not graph editor. It's a separate window module, and even has it's own category in input panel of user preferences. Blender's timeline is very poor-featured and does not even allow for selection of keyframes within it, so left mouse button will not interfere with absolutely anything.
2, When it comes to graph editor, introducing LMB scrub may result into interference with Blender's LMB tweak feature, but graph editor has completely different, two dimensional axis scrubbing, unrelated to timeline. It can be handled as follow
A, When select with is set to LMB, RMB will still scrub the 2D crosshair in graph editor when clicked anywhere in the graph area.
B, LMB will still perform tweak
C, LMB over the scrubbing cursor would act in the same way viewport transform gizmos do. The hotspot around the cursor lines would be derived from the hotspot value under manipulator toggle in Interface Panel of User preferences.
D, It would be appropriate to change the cursor appearance to indicate the cursor is hovering over scrubbing crosshair, turning it into simple cross (+) icon would help a lot. A same behavior should be also introduced for View 3D gizmos, to give user feedback if cursor is hovering over the gizmo hotspot.

But back to original point. This request is not invalid. Timeline is not Graph Editor, and timeline does not offer any possibility of selection of frames within it, therefore using RMB to scrub it, especially when interaction mode is set to LMB, goes against common sense.

EDIT:
Just noticed blender offers some form of selection in timeline, using markers as you said. But markers seem to be very fringe, rarely used feature. A solution in this case is trivial. LMB should scrub the timeline as long as there's no marker under a cursor. When cursor is hovering over a marker, marker should be scrubbed instead of frame cursor. Two things are crucial here:
A, Marker's hitbox for cursor should be small enough, ideally contained only in bottom half of the timeline, so that you can click above the marker and start scrubbing frame cursor even if marker is in that location.
B, In the same spirit as mentioned above, cursor appearance should change when cursor is hovering over marker's hitbox.

One more thing. I think that the cursor icon that appears when hovering over viewport customization corners, a cross icon, would be ideal for this.

The timeline is one of multiple editors that has time along the X axis, like the sequencer or graph editor. We try to be consistent in the UI, so scrubbing should work the same in all editors that support it. And there does exist marker selection the timeline which this conflicts with, and maybe keyframe selection in the future.

Making scrubbing work like tweak would make it slower to use, currently you can just click anywhere rather than dragging, and if the current frame is outside of the visible range it would be even more complicated to use. Further in the graph editor you often need to edit keyframes at the current frame, so having the cursor lines take priority over that would be a problem.

By the way, the reason it's marked invalid is because we do not classify this as a bug. It's a valid concern, but this is a UI design discussion, which we do not handle as bugs in the tracker.