CI is not your answer, but could be helpful. What you do right now is really good. You shouldn't compile your stuff locally and push that in your repository when pulling it. Your server is capable enough to compile everything down and run it.
However you have a lot of servers running and letting all of them compile the assets is a waste of time. This would be fine with one or two servers, but with more it's ideal!
The best solution for this is using some kind of CDN. So another website or domain that will serve your files. This way you only have to compile them once and your other servers can then pull the data from there.
You can Google around on how to do this in Laravel. There are even packages for this that connect with third-party CDN companies with an API so your assets are always available.
Another solution would be to use containers. With containers you can work locally with the same system and assets as you would on production. Your local development container can even be deployed to productie and would work just fine ;)