Are you sure the Mendix Cloud is expensive, or is the Mendix license expensive? You will require a Mendix license to run your app in anything but the Free Tier.
I believe Mendix doesn't make a – or much - profit on the cloud environments: that's not their business model. You can, however, deploy on Dockers or on VM's in another cloud: I've hosted apps in Google Cloud Hosting and in Docker in AWS Fargate. See documentation on deployment option here.
You can deploy your app to any Cloud platform as a private cloud. Use EKS or AKS or Openshift platform. still, you have to provide resources like Database or Object Storage