The company MakeMHz has just announced a true digital-to-digital HDMI mod for the original Xbox. This is a no-cut mod that replaces the analog output port with an HDMI jack. Audio is 100% digital as well, making this the cleanest output solution currently available for the Xbox. Pre-orders are open now for an introductory price of $60. The price will go back up to $80 after pre-orders are over: https://makemhz.com/products/xboxhdmi
The output is 1:1 of the original signal, which will look awesome when the 50-ish 720p-compatible games are played. Unfortunately, there’s no scaler or deinterlacer, so the look of 480p games will depend on your TV and if your favorite game is one of the 20-ish that only runs in 480i, it’ll be a pretty laggy experience on most TV’s (maybe just force 480p mode with a mod chip?): https://en.everybodywiki.com/List_of_Xbox_games_with_alternate_display_modes
Since this removes the analog port, that means dual-output will require some external devices. You should be able to use an HDMI splitter and a basic digital to analog converter, if you’d like to send one output to a CRT and another to a capture card. I’m hoping to get one of these in for review and plan on testing every aspect of it.