No the software center / solus-sc does not need to updated for the third party section or the actually repository to work properly it is independent of those sort of changes. I.e it was last updated 2 months ago and that was to "Remove HTML comments from package changelogs"
Are you saying when you open the software center it closes straight away or when checking for an update to TeamViewer it closes after awhile or after click check for updates it appear to be doing something and then stops?
You can check the installed version of TeamViewer by using eopkg info teamviewer
the latest version is 15.48.4 I updated it to this version 5 days ago.