Rewards button pre-opt-in badge dismisses after first button press (uplift to 0.73.x)

[brave-builds]

Uplift of #3898
Fixes brave/brave-browser#6691

Approved, please ensure that before merging:

• You have checked CI and the builds, lint, and tests all pass or are not related to your PR.
• You have tested your change on Nightly.
• The PR milestones match the branch they are landing to.

After you merge:

• The associated issue milestone is set to the smallest version that the changes is landed on.

[kjozwiak]

Restarted macOS due to the following failure as per https://ci.brave.com/job/brave-browser-build-pr/job/pr3898_rewards-stub-dismiss-badge_0.73.x/2/execution/node/665/log/:

16:32:02  In file included from ../../headless/lib/utility/headless_content_utility_client.cc:16:
16:32:02  ../../components/services/pdf_compositor/pdf_compositor_impl.h:21:10: fatal error: 'components/services/pdf_compositor/public/mojom/pdf_compositor.mojom.h' file not found
16:32:02  #include "components/services/pdf_compositor/public/mojom/pdf_compositor.mojom.h"
16:32:02           ^~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
16:32:02  1 error generated.
16:32:03  [24083/38677] CXX obj/media/learning/mojo/public/mojom/mojom_blink/learning_task_controller.mojom-blink-test-utils.o
16:32:03  [24084/38677] CXX obj/media/learning/mojo/public/mojom/mojom_blink/learning_task_controller.mojom-blink.o
16:32:03  [24085/38677] CXX obj/media/learning/mojo/public/mojom/mojom_blink/learning_types.mojom-blink.o
16:32:04  [24086/38677] CXX obj/media/midi/mojo_blink/midi_service.mojom-blink-test-utils.o
16:32:04  [24087/38677] CXX obj/headless/headless_shared_sources/types_network.o
16:32:04  [24088/38677] CXX obj/headless/headless_shared_sources/protocol.o
16:32:05  [24089/38677] CXX obj/media/midi/mojo_blink/midi_service.mojom-blink.o
16:32:05  [24090/38677] CXX obj/headless/headless_shared_sources/types_page.o
16:32:06  [24091/38677] CXX obj/headless/headless_renderer/headless_print_render_frame_helper_delegate.o
16:32:06  [24092/38677] CXX obj/headless/headless_renderer/headless_content_renderer_client.o
16:32:13  [24093/38677] CXX obj/headless/headless_renderer/headless_content_main_delegate.o
16:32:13  [24094/38677] OBJCXX obj/headless/headless_renderer/headless_content_main_delegate_mac.o
16:32:21  [24095/38677] CXX obj/headless/headless_renderer/headless_web_contents_impl.o
16:32:21  ninja: build stopped: subcommand failed.

[kjozwiak]

Uplift into 0.73.x approved after deliberating with @brave/uplift-approvers. This fix was also uplifted into 0.71.x as per #3914 and will need to be uplifted into the remaining channels. Both CI/Jenkins and CI/Travis passed after several restarts.