A bit old topic. Still current problem for me.
"The issue is this: after some time, Outlook Client refuses to send email to Google Mail (Gmail). There doesn't seem to be any particular reason for this happening... " My solution is to go straight on Gmail Web and not use the desktop app.
I use Office Pro with Outlook 2016 client (home) on W10 Pro with IMAP on a well running machine i7 1TB SSD on fast ethernet connections. But I use it frequently on 2 other W10 Pro machines with same experience.
I do not count the number of times where I deleted the gmail account in Outlook to get it recreated. Which works well & fast, ... for a while. I never encounter such problems with hotmail account (Outlook in Cloud). Outlook Cloud is a robust & powerful product making me rethink my use of it. Historically (I am quite old) I used hotmail since 1997 for all "commercial" as it had better spam filters well before gmail, and gmail since 2004 for all more "private" emails. I abandoned Compuserve and all ISP related mail accounts to work only in IMAP.
Still today, same problem as described above.
To be more precise I create a message in Outlook client (a little text with 3 little attachments), it goes in Outbox folder to be sent with gmail account, and it just stays there for some reason. I delete it, recreate the message with same effect.
A bit fed up of trying to find a root cause or to go round the problem by recreating the client folder.
Hence my search today, considering to get rid of Outlook client and discovering here a hint that the root cause could well be somewhere else: it might be that the problem is with the gmail server & not with the eM or Outlook clients.
Other context where I have sometimes (not always) sync or comm issues: moving emails between "labels" / "folders", deleting large chunks of Gmail emails in the client, reorganising in the client, archiving in the client, sending an email with a e-signature certificate and storing it in a different label, sync of labels <-> folders: this sync table gets corrupt. The difficulty of matching responses (S/MIME receipts) between client & server can block the client. Solution then is to delete completely (including in trash) the email "suspected" to create the blockage. When lucky, only then the sync resumes for other messages to flow back and forth, otherwise all seems blocked. So, sync is good ... when it works.
It appears to me that Gmail works with labels attributes, not hierarchical folders. From my limited experience, Outlook synchronisation between "folders" & "labels" seems intrinsically complex or technically doomed. I just read that eM seems to have mirrored this gmail architectural approach from realease 7. But it still seems to have its own issues with this sync or comm.
I need a desktop mail database because I often work in areas with no internet or radio network at all. I still need to retrieve things and to prepare mails. I also need to archive old stuff (each year at least) to get light & swift cloud mail accounts. I need to access these archives easily. Outlook manages this with ease.
Gmail Cloud is great and I use it a lot on Android as very easy to use. So, solution for me might well be to still use Gmail but forget about using "labels":
so to paraphrase again "set it, use it for much simpler use and forget it." Indeed, a lead to be further tested.