The slack attachment format defines a fallback field, that is required for attachments to be valid, and is used when the full-formatted message is not usable in a particular context (e.g. text-mode clients, notifications).

For example, Slack uses that fallback field for the text of notifications. I believe that Mattermost should do the same.

That would require changes on the webhook-related code, on the format of the post object that is forwarded in Javascript, and on the code that generates the text shown in notifications. I could work on a PR if you are favorable to the idea.