Is WP Pusher the best option for handling theme updates across many sites?

My company is working on a custom Wordpress theme that is installed across several of our client sites. We are trying to figure out the best way to handle deploying theme updates across all the sites. We don't have a staging environment for each client, so I don't think we want a GitHub commit to automatically trigger deployment. I would rather have control over which individual sites get the update. I see that the WP Pusher does the sort of thing we need, but I wanted to know if there are any other good options. Price is going to be a factor, so something cheaper or self-hosted would be great. Most of our sites are currently hosted by AWS Lightsail instances.