SSDT is no longer needed. To build SSIS packages you should install Visual Studio (you can use Community if you don't have a license for a full version). NOTE: You must currently use VS 2019, not the newer VS 2022 as SSIS is not yet supported on the newer version. After you install VS 2019 then install the SQL Server Integration Services Projects extension.
You do not need to have VS installed on the SQL machine. SSIS runs on your SSIS server (generally your SQL server) and therefore you just need to deploy the package to that server. It is not recommended that you install development tools on a production server under any circumstances. You can debug your SSIS package locally on your dev machine in VS.
Do be aware that SSIS is tied to the version of SQL you'll run it against so ensure that your SSIS packages are targeting the version of SQL you need. If you need an older version of SSIS then you might need to also run an older version of SSDT as SSDT is tied to specific SSIS versions.