Can a 2001 Honda Shadow VT750CD ride without a front fender?

I removed the front wheel to install new pads and to tighten down the front fender because it was rattling on bumps. The 4 fender/fender brace were extremely rusted and the heads all broke off. I now have a drillbit and a broken screw extractor broken off in them. While I figure out how to get all of these removed and retap those threads, I'd like to still ride. When I put the front wheel back on, the brake rotor is now rubbing on the brake caliper metal. I have seen people remove their fenders before on bikes, I'm just not sure if it works on mine. With the fender and fender brace, perhaps it pushes the forks away from eachother? Are they necessary on my honda shadow, or did I install the front wheel/front caliper incorrectly and it should work fine fenderless? Can a 2001 Honda Shadow VT750CD ride without a front fender

The fender is not structural but its bracketry does help brace the forks and reduce twisting.
The rubbing problem is more likely to be down to the spacers etc not being fitted correctly when you put the wheel back in.

You put it back together wrong and yes, you can ride without a fender. Don't do it until you fix the brake

Just wait till you hit a puddle or some gravel on the road.

Yes the front fender does nothing that would stop the bike from operating. You will notice some water being sprayed on your when you drive on wet pavement.