I would not install the tools on a server - those tools are meant to be installed on a workstation (desktop). With that said, yes - you need to install Visual Studio 2019.
There is no SSDT-BI for VS 2019 - to get those you add them as extensions. During the installation of VS 2019 - make sure you select the 'Data storage and processing' tools under the heading 'Other Toolsets'.
After installation - use the option in VS to manage extensions. In that dialog - select Online and find the extension for Reporting Services and download. Then find and select the extension for Integration Services. The Integration Services extension will be placed in your downloads folder - and must be run separately. The Reporting Services extension will be installed when you close VS.
The order of operations is:
- Install Visual Studio 2019
- Manage Extensions - selecting those you want
- Close VS - extensions will be applied that can be
- Once the extensions have been installed - run the Integration Services extension install
Once you have that done - you can launch VS, create a new project from a template - select the Reporting Services template and start developing the report.