I'm almost certain there was a DOS menu program with joystick support somewhere, but I can't remember the name. In any case, that would probably be the way to go: configure all games in that menu app, add it to autoexec and use joystick to navigate from couch either using a menu program or some TSR that converts joystick action into keypresses (unless that would conflict with actual games).
For video I had nice results with Gefen VGA2HDMI scalers, it does video as well as sound (up to 1080p60, if I remember it well).
The biggest question would be just how playable the games would be with a joystick. I mean, shooters would definitely be a no-go (IMHO not even some of the modern "boomer" shooters work well without a mouse). Flight/Space sims often require a lot more buttons than a regular DOS joystick/gamepad can provide (unless we're talking about HOTAS, but even then a keyboard might still be required). That would probably leave platformers (think Jazz Jackrabbit, Duke Nukem 1/2, Jill of the Jungle, Commander Keen) and maybe most driving games.
Then, of course, there is a problem that there are no good wireless joysticks for DOS machines. I think there are dongles that allow connecting modern gamepads to Gameport, but I don't think those work with bluetooth or 2.4g joysticks.
All in all, it would be a very complicated setup with ultimately not a great experience. IMHO a better option would be getting something like Raspberry Pi or a cheap x86 box, installing some kind of launcher (EmulationStation, Pegasus, etc) and getting source ports. There are great open source ports for a whole lot of DOS games, from Doom/Duke Nukem/Quake to Tomb Raider, Jazz Jackrabbit, Descent and even XCOM. Most of them have modern gamepad support and extra features like better rendering options, MIDI soundfont support, etc.