We also don't allow SSH on our servers at LittleBizzy, it only welcomes problems. That being said, I personally don't find WP-CLI any easier/faster to use than your basic shell.
While I can appreciate being stuck on SSH, I think a lot of developers treat WordPress projects just like they treat any other project (Magento, Python, localhost, whatever it may be).
In short, treat WordPress as WordPress and learn to appreciate things like "plugins"... after all, these things are the reason why WordPress is the #1 CMS in the world!
WP Migrate Pro or All In One Migration can both rapidly move WP Engine (or any sites) to another.
As @leland touched on, Git also helps the super nerds; check out VersionPress too.