TL;DR: Doing a full restore of iOS 14.2 through iTunes (not OTA) still doesn't solve it, at least not when also restoring from a backup.
Some of you may have already discovered this in your internet travels, but this is actually a pretty longstanding issue. If you do searches on the terms: "spinning icon" or "spinning network activity indicator" along with either "dictation" or "speech-to-text", you'll see this goes back several years. Although there are a lot of posts about the spinning icon, we're not the first group to discover that it is tied to the dictation services. People were also posting about the temporary Siri fix several years ago.
Back in 2017, someone was successful by doing a full restore through iTunes, versus OTA. I know some folks in this thread have tried resets and restores, but I don't think anyone stated they had tried that, so I went ahead and gave it a shot. I completed a backup through iTunes first and then did a complete reinstall of iOS 14.2. I then restored by backup. This whole process takes quite awhile and I can see why no one wants to do it.
After completing the above, I initially thought it was successful. I did a couple of short dictations and there were no issues. I was ecstatic and thought it had worked, because I received the persistent spinning icon 100% previously. So I went on my merry way finishing with the restore (logging into accounts, checking settings on apps, etc.). When I went back and did another test dictation, the **** icon came back and kept spinning.
At that point, I went ahead and enabled Siri. That band-aid approach does seem to be working for me also. It looks like I don't even have to say anything, just activate Siri with the home button and let it sit open for a few seconds and then close it and the persistent spinning network activity indicator goes away.
Between the apparent initial success after the restore, and then discovering the problem actually still persisted, I had done very few things. I had disabled Clips and Game Center from backing up to iCloud (I'm not sure why they were enabled since everything else is disabled), I adjusted some display settings (I think I had to re-enable the Zoom setting), and I signed into my Google account in the Google Calendar app. I'm pretty sure that is all I had done.
I know I more complete test would be to do a clean install of iOS 14 and then set the phone up as new, versus restoring from a backup, but then the testing process after that would take days. I'd have to set up as new and then check for the problem after every single app install, settings tweak, account sign in, etc. I'm just not willing to do that.
I also looked into downgrading the iOS, but 14.2 is the only signed version right now, and it looks like trying to downgrade to an unsigned version is pretty involved.
I no longer have hopes that Apple is going to address this problem though. It has been a known issue for a long time, just do the search I suggested and see for yourself. For a product that prides itself on "it just works", I'm pretty underwhelmed.
Good luck to all!