Is this a thing that is possible? In other words to create a template in Mosaico (that is, the Mosaico CiviCRM extension) and to store it for use in User driven message templates such as emails?
You can try the Mosaico Message Template Synchronisation extension.
From the README:
This extension enables you to use CiviCRM-Mosaico templates with scheduled reminders, personal messages, etc. It does this by automatically copying each template from Mosaico to a CiviCRM "Message Template".
When you install the extension, all existing MosaicoTemplates will be automatically copied to MessageTemplates.
When you create or update a MosaicoTemplate, the corresponding MessageTemplate will be updated.
There's a few gotchas/caveats to look out for - see README on branch issue/6 for fuller explanation about what happens with things like unsusbscribe links which don't apply to a log of message templates.
Under Mailings, you should have an option for "Mosaico Templates" where you would create your message templates, but these templates are currently only available to you for mass email (Mailings > New Mailing) - they can't be used for things like Schedule Reminders or direct emails to contacts using the Send Email action (to 50 contacts or less) in CiviCRM.
Sorry not to be able to tell you something nicer, Tamar