Chances are, it has nothing to do with your problem – unless you already deleted client IDs from Google Dev Console even though you have apps with GPGS that are linked to the deleted client ID.
I’m not sure what kind of errors it would cause by deleting a project/clientID that is linked to apps in GPGS.
You mentioned you already have a published app. I certainly wouldn’t delete a project/clientID that are linked to your published app.
I would only delete projects/clientIDs that are linked to unpublished apps in Google Play Dev Console.
Basically, I wouldn’t start deleting things from Google Dev Console unless I know exactly which projects/clientIDs are being used and not used. If a project/clientID is linked to any of my published apps in GPGS, I wouldn’t delete it.
And if you’ve already deleted one by mistake, you can undelete it within 2 weeks or so after the deletion.
If your Google Dev Console have many projects that were inadvertently created and currently doing nothing, you may just want to leave them be for now.
If you have many projects with same name/title, you can change it so that you can easily tell which project is which. And then, eventually, you can delete the ones that aren’t doing anything. (The reason why I wouldn’t delete projects right away is to make sure projects-that-shouldn’t-be-deleted don’t end up being deleted. Having some unused projects sitting around won’t do you any harm. You can delete it when you are ready to clean and organize – but when things are not quite working right, it might be better to wait.)
Naomi
Edit: By the way, when you mentioned “console”, I assumed you mean Google Dev Console (not Google Play Dev Console.)