Here is one of the fatals error :
PHP Fatal error: Uncaught Error: Class "WPML_Cache_Directory" not found in
The only way i can make my site work now, is to deactivate all my plugins, by database query. Re-install all my plugins, but not wpml related.
Delete all wpml related plugins.
Install again OTS Installer. And Install by OTS, WPML and all the other subplugins.
As far as I understand, the code for our plugins reaching your site should be of the same kind as the code from any other plugin. There is no reason for it to behave differently than any of your other plugins. No matter if you use OTGS Installer, or you upload a ZIP and unzip it, or if you use the WordPress admin interface to upload the ZIP over there. Your workflow using Git should work the same - counting that WPML might be a bigger plugin than others, and contain more files.
That being said, I think that there is no action that we can take on our side to ease your process. The nature of our plugin is the same as the nature of any other WordPress plugin. So the only thing that I can imagine is a limitation of the number of files that you can move at once, or a timeout cloning or copying changes, or a caching problem. But those are only educated guesses.