I see it ambivalent using an own browser for debugging purposes. It would be better if IntelliJ IDEA respectively the debugger isn't dependent on any kind of (own) browser. People use different versions of browsers. Those browser properties and behaviours are different and must be tested. Beside that it is not clear on which standard/version the own IDEA browser is based. Thus it should be possible to use and connect different versions of one browser installed outside IntelliJ IDEA. The browser and its version should be decoupled from the debugging facility itself.
Some ideas on a possible solution: The user/developer can start the web application e.g. via context menu "start with" > (a) Internet Explorer (b) FireFox (c) xyz respectively with the standard browser on the individual desktop - and the debugger works (while using one of those browsers). Some kind of plugin architecture could be helpful.