MihailMate pointed out a solution to this problem. I want to rephrase that solution here to make it a little more clear what to do.
- Open System Preferences, and then click on the Dock & Menu Bar icon.
- In the left-side panel, click on Do Not Disturb.
- On the right side of the window, mark the check box for Show in Menu Bar. You should probably also select the "always" option in the drop-down box below that check box.
This will cause the Do Not Disturb icon to appear in the system menu bar. If you don't like its placement in the bar, Command-click and drag the icon to a different location in the bar. Unfortunately, this workaround requires you to clutter your menu bar by adding an extra icon which you may not otherwise want there.
For me, this seems to prevent the clock text from becoming dim even when Do Not Disturb mode is on. On my computer, I have set Do Not Disturb mode to turn on automatically from 10:10 PM to 10:00 PM (via System Preferences > Notifications > Do Not Disturb). This is kind of a hacky workaround to force Do Not Disturb to be permanently off. So, I will need to wait until tomorrow to see if the clock text remains fully visible even overnight after the system toggles Do Not Disturb mode off and then back on again.
I work as a designer, and I find it incredibly stupid that the brightness of the clock text in the menu bar is dependent on the state of the system's Do Not Disturb setting. Those two settings have nothing to do with each other. Somebody at Apple was not right in the head when they designed that change.
This glaring flaw is one of the worst changes to Big Sur, which I feel is an abomination of an operating system, from a design perspective. Personally, I see this as a sign that Apple's design team has been going down the toilet for quite some time. I froze all of my personal-use Macs at Mojave and I only installed Big Sur on my spare Macbook so that I can know what the user experience is for the majority of users.
Speaking of macOS versions, Catalina is not terrible, except for the Music app which has a terribly crippled UI/UX compared to the previous version of iTunes in Mojave. It's unfortunate that you can't upgrade to Catalina without also "upgrading" iTunes to the new Music app. If Apple's design team was smart (which clearly is no longer the case), they would decouple their suite of apps from the core operating system. Thus, you would be able to upgrade the OS without also being forced to upgrade all of the Apple apps like Pages, Contacts, Reminders, iTunes/Music, Calendar, etc. Users should be given the option to upgrade those apps independently of the OS. Often, design changes to those apps are not desirable, while users may still want to upgrade the core OS. Although in the case of Big Sur's design changes, I would say to avoid that like a computer virus.
Hopefully, Apple will eventually get their head screwed on straight in the design department. As a consumer, I have so many Apple devices that it would be a hassle to switch over to a different ecosystem. But I'm starting to wonder more and more about the world of Windows (or maybe even Linux) and curious to see what the user experience is like over there these days. Maybe it has become better than the current state of the Apple way.