Hi,
Per my knowledge, the download of .rdp file is required to launch the connection of remote app. The workaround you've used should be a good idea to avoid users manually saving the file but open the app directly.
Just as you mentioned, only IE will not download .rpd files since it supports the Remote Desktop ActiveX control but others don't.
Here is a similar thread which explains the ActiveX with more details:
https://stackoverflow.com/questions/31000445/using-ts-remoteapps-on-chrome-firefox-without-downloading-rdp-file
Please note: Information posted in the given link is hosted by a third party. Microsoft does not guarantee the accuracy and effectiveness of information.
Hope this clarifies and please help to accept as Answer if the response is useful.
Thanks,
Jenny