To get ASIO working the software output that you're trying to capture needs to have an ASIO output and that output would have to be set to JRiver's ASIO line-in input. Most windows software does not have an ASIO output, and Netflix in a browser defnitely doesn't output in ASIO, and can't ordinarily be configured to do so without some third party software (like the ASIO bridge that some folks on the forum use).
Whether or not you use ASIO line-in as an input to JRiver is unrelated to whether you're using JRiver to output to your soundcard using ASIO.
WASAPI loopback, by contrast, just intercepts whatever is being sent to the windows default sound driver, which includes netflix in a browser. JRiver grabs it and sends it your sound card using your output mode of choice (including ASIO).
Basically if you need ASIO input to work, look into ASIO bridge software, but all that software is doing is intercepting the non-ASIO output from Netflix and re-packaging it as an ASIO stream. The main reasons to do that would be if your system doesn't work well with WASAPI loopback (for some reason), or if you don't have an extra on-board sound device.