You upgraded your SharePoint 2010 farm directly to SharePoint 2016 with some help from your friends at Sharegate? Like Ben said: that’s cool!
But your organization is still using Office 2010 because your boss don’t understand why you need a new Office version, because you can also type a letter in Word 2010?
Convince your boss this is a bad idea. With Word 2010 it’s possible that your document in SharePoint 2016 is getting corrupt, locked or co-authoring is not working the right way. Check your uploadcenter for all the files which has a problem with uploading.
The good news is that some things with Office 2010 and invalid documents are fixed in the January 2018 update of SharePoint 2016 but Co-authoring is working a different way now a days, so I don’t think it solves all your cobalt errors in the event viewer but let’s give it a try next week!