Fix race condition in Brave Rewards panel that results in inconsistent UI ("Your wallet" vs verify wallet)

[Miyayes]

Description

In the Brave Rewards dropdown panel, sometimes the panel will display "Your wallet" text, and sometimes it will display "Verify wallet" (or "Wallet verified"). It is inconsistent, and according to some investigation, is due to a race condition in the code.

In no case should it show the "Your wallet" text. The panel should always show "Verify wallet" (or "Verified wallet" if the user is already verified). The "Your wallet" text should be removed, and the race condition should be fixed.

Your wallet case (incorrect):

Verify wallet case (correct):

Steps to Reproduce

1. Open the Brave Rewards panel
2. Observe that sometimes it shows one or the other

Reproduces how often:

Depends on race condition.

Version

Happens as late as v1.19.

[emerick]

The onboarding modal is also not displayed when this occurs.

[LaurenWags]

Verified passed with

Brave | 1.23.25 Chromium: 89.0.4389.86 (Official Build) nightly (x86_64)
-- | --
Revision | 0c2dac31bc21e806a791641316e99920f84ea52c-refs/branch-heads/4389@{#1472}
OS | macOS Version 10.15.7 (Build 19H524)

Verified test plan from brave/brave-core#8113.
Verified STR from description.
Tried test plan steps ~15x to reproduce the issue using 1.21.74 and was unable to reproduce.
Tried test plan steps ~15x on 1.23.x and did not see the race condition.