Issue / Problem
I faced this issue when working with a “radix” theme on a druapl 10.3.2 website. Upon inspecting the status message block, instead of desired outcome shown in the green rectangle in the screenshot; that is: with hook suggestion and override twig template name suggestion). I’m getting what’s shown in the red rectange.

The issue issue as “prashant.c” described in issue 3456176 - #10:

Mitigation/Solution 1 - Theme Hook
The first solution is proposed in #17 comment under the issue, via creating a theme hook to control the rendering of the system message block:
In your yourThemeName.theme file, add the following:
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Mitigation/Solution 2 - BigPipeline
The second solution is proposed in #45 comment under the issue, via bypassing some portion of the code in the core/modules/big_pipe/src/Render/BigPipe.php file:
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Outcome
Both approaches worked for me, and as a result I am able to create my own custom status-message.html.twig file my radix sub theme to customize how the system message get displayed:
