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Disclaimer: I do not want to request access to your video, so I did not watch it. If it contains important points, please make it readily available. Posting screenshots here could be useful, too, if you're able.

I'm not sure what you mean by "cumulative costs," but the global planner does use all of the costmap (total cost of its layers). Once a planner finds an acceptable path, its job is done unless you ask it to try again. That can happen based on timing or, as in your case, when the planned route becomes impossible.

The first thing to inspect is your move_base configuration file (move_base_params.yaml maybe). There is a planner_frequency parameter, which dictates how often in Hz the move_base node calls the global planner to retry. You may also be interested in recovery_behaviors, which controls how the robot tries to get itself out of trouble (can't find a path at all). The move_base documentation provides more details.

Although I answered your question as stated, I know what you're really after is functional navigation. If the above parameters are set appropriately, then move_base will call the planner until it has exhausted its options. Does the robot perform its recovery behaviors? Are you sure that the "alternate path" is viable and that the planner is able to see and evaluate it?