Sdr, need to take one or two steps back ,
All the Web Sdr provides is a remote tuned radio reciever yes some have on site decoding, but , that's really a fix for modes that cannot self synchronise , and suffers from web delays,
All the Web Sdr is providing, is the recovered ''audio'' stream, which in turn, is presented to the decoding software, just the same as any sound card based data mode , the audio recovery is probably taking place in the application running on the local ''pc'' ( there are browser compatibility issues - audio ) the Web load is reduced , the waterfall is a graphic display , the only section of spectrum conveyed , is determined by the locally selected bandwidth .
Using the audio mixer facility , in say win10, the recovered audio from the Web browser , may be routed to a wave recorder , data mode, speakers etc
..your right, live path is the best test , a well located Web Sdr, should produce sufficient ''test data'', you can loop