Your API in question looks good, seems it just call the download url of entire drop folder. Just like this ticket: How to get the download url of the artifact in Visual Studio Team Services?
According to your description, even through the browser it still keeps giving credential pop-up.
If so, your issue may related credential cache. Try to clear any TFS credential cache stored in Control Pane--Credential Manager.
Besides, instead of using account/password, you could also try to use PAT. Username should be blank. PAT is the password.
Since you are using PAT with on-premise TFS server, check if you already turned off Basic Authentication.
We recommend you keep IIS Basic Authentication turned off at all times
when using Azure DevOps Server. Only if necessary should you enable
IIS Basic Authentication. When IIS Basic Authentication is enabled on
your windows machine, it prevents you from using personal access
tokens (PATs) as an authentication mechanism.
Source link.
About more details for how to use powershell to download build artifacts, you could also take a look at this blog-- Downloading Build Artifacts in TFS Build
