Installing .NET Framework 3.5 resolved the issue for me.

My old server had SSRS version 13.0.5882.1 on Windows Server 2012 R2 Standard. My new server has SSRS version 16.0.1113.11 on Windows Server 2022 Standard. After hours of troubleshooting the only difference I found was the old server had both .NET Framework 3.5 and .NET Framework 4.5 installed, whereas my new server only had .NET Framework 4.5 installed. After installing .NET Framework 3.5 the barcodes started generating again.