Taking your questions one by one:
DXA web application cannot publish to filesystem. Is that right?
In general, the pages served by a DXA application are not "published to the file system", however binary assets can be published to Tridion Content Delivery, and are then available via the services. This is how your configuration settings are managed, and also your front-end assets if you choose to use the DXA for those. In principle, your application can do anything you want with the data it gets from Content Delivery. Furthermore, you aren't limited to what DXA can do. Content Delivery can still deploy to a file system, but this is deprecated in 9.5. You'll have to use judgement to determine what's a good idea and what isn't.
Daily management is very happy with current (no DXA) publishing. How can we make them happy again?
They will be happy as long as you deliver reliable systems that do what is necessary. Updating to more modern architectures is part of that. If you don't do it, then it will become progressively more difficult to keep them happy, as various components of your system are no longer supported and so forth.
How are we going to solve this as long as there is no continuous deployment process in the company?
Try not to let continuous deployment be a barrier to progress. You already have a system that can accept a zip file and deploy your application. Using Tridion for this deployment in the context of DXA wouldn't win any beauty prizes, but it would be possible. Using Tridion would make it needlessly complex compared with say, file system replication or similar techniques. Even then, you'd probably be better off containerising your application, even if the deployment process is manual. Don't let the perfect be the enemy of the good. Make small steps in the right direction.