(post has been rephrased due to later findings)
Hello fellow community members,
First of all, it is not an easy issue, I have spent litterally WEEKS on this issue by my own and with several support tickets.
I even was forwarded to the Office "Technical support" level, who reviewed the chat session I had with level 1 support and just closed the session "as is", kicking me out (thanks Karl, I saw your name when you wanted to write something you didn't send) : I know it's not an easy issue but it's not a way to treat a client. I'm honestly doing my best to apply the guidance again and again, even from scratch.
NB: The issue impacts the M365 Office installed applications, as Web versions work as expected
(new user account / working fine <=> administrator account
current user account / problematic <=> normal user account)
Since I bought the Copilot Pro subscription recently, the most straightforward symptom is :
With the same M365 account :
- Copilot appears and can be used when starting Word as a new user account (O365HomePremRetail = vNext)
- Copilot does NOT appear when starting Word as my usual user account (O365HomePremRetail = Legacy)
Here is a screen when starting Word "as new user account" :

The lesser straightforward symptom is :
- when I start Word as the problematic user, even if I'm already authenticated (profile with name is visible in the top-right corner) I get a popup suggesting me to authenticate (screenshot shows an account is connected, and Copilot does not appear near "Components")

And after authenticating successfully with the same M365 account, the error code 29 appears :

(going through the "connect to check your subscription" process or not does not change how Word behaves)
List of what was verified/tested, by myself and by support, unsorted :
- The Microsoft 365 account I use has a currently valid Family licence

- M365 apps are installed via the setup listed on the M365 subscription page ("install" button that provides OfficeSetup.exe SHA-1 sum is B15AE8A2E60F811EAFA580AE49FFE6AFD81C5458)
NB : I tried to install it directly using a fully authenticated administrator account, and another time using the normal user account + accepting the admin elevation when asked

- Before installation I successfully removed all Office/M365 apps from the computer, several uninstall/reinstall tries, uninstalling them via control panel and via the MS Support / Fix tool to remove all Office components (https://aka.ms/SaRA-OfficeUninstallFromPC that gives SetupProd_OffScrub.exe, SHA1 A236FFD2C90D5F4E9CEBBC53FA60B17949950CC8, complete cleanup process took 30 minutes ~) : there was no more "Office" / "Microsoft 365" installations when I installed via OfficeSetup.exe

- Computer is running Windows 11, 23H2 with all updates 22631.3085, and Microsoft 365 apps / Word shows version 2312 (build 17126.20132 Office "Démarrer en un clic")
- I already tried using the Microsoft Support Fix/Repair tool https://aka.ms/SARA-OfficeActivation-OF that gives SetupProd_Act.exe, SHA-1 B3B53C99EBE9A8D9F5416403FC7B7D2F1703CB98, to try to fix any licencing issue, the tool run "successfully" but did not fix the issue
- I tried disconnecting / reconnecting the Microsoft 365 account from within Word : when connecting for the first time after a fresh install it correctly shows a message telling that the licence has been correctly activated
- Here is a screenshot of the "Account" screen when starting Word as administrator (+ I was clicking on the profile button to verify the account) :

- Here is a screenshot of the "Account" screen when starting Word as normal user :

- I compared which application was effectively being run, using ProcExp (Sysinternals), when running as normal user and as admin :

(when running ProcExp as admin, the popup balloon doesn't show on screenshot, but Command Lines are the same than when running as normal user)
- I initially used a Windows 11 LOCAL ACCOUNT as normal user, and I willingly converted it to a Microsoft Account using the "Connect" feature, but using a Microsoft Account did not fix the issue, I tried to get back to a Local Account to see if it changed anything, but it did not, so I'll get back to converting it to a Microsoft Account again
(==> THIS may have initially caused the issue when I converted the local account to a microsoft account, as I was NOT having the issue before <==)

- I used OSPP.VBS /dstatus and /unpkey to remove any unused licence activations (as guided by support level 1), the only remaining one is the M365 subscription :

(exact same information when running OSPP.VBS as normal user / as administrator)
- I tried using OSPPREARM.EXE (I took the initiative myself to test it), the result was that the very next time I ran Word there was no more "check your license again" screen, but Copilot was still NOT visible, then I closed Word and started it again, the popup reappeared (= back to the same state)

NB: when trying to rearm again an error message says that I need to reinstall my operating system to be able to rearm again
- I used vNextDiag.ps1 to display license status (I found this by myself, support never suggested using this tool) :
Running as normal user :

Running as administrator (THERE IS SOMETHING DIFFERENT HERE THAT MAY BE A CLUE BUT I DON'T KNOW WHAT TO DO WITH IT YET) :
==> I'm currently investigating this specific clue as "missing vnext license" / how to switch from legacy to vNext for o365homepremretail, etc., I have no clue what this means yet<==

My current exploration step is
https://learn.microsoft.com/en-us/office/troubleshoot/activation/reset-office-365-proplus-activation-state
"You might need to perform tasks such as the following for your organization:
- Tenant to tenant migration
- Repurpose a device for a different user
- Change the license mode for Microsoft 365 on a device"
I'll get back to update the post if I find more information..
If you already know how to jump to a potential solution, feel free to post, but be sure to read the post entirely to see which steps I already tried :)