How necessary are barrel adjusters? In the process of getting the slack out of my rear brake cable, the "cap" on the insulated part of the cable pulled out of the barrel adjuster. While monkeying around trying to get that part of the cable to "catch" back inside, I backed the barrel adjuster all the way out of the handle and can't seem to get it back in--I may have stripped the threads. The brake is working fine; I went for a short ride and had no issues with it. My concern is that not having the cable seated inside the barrel adjuster long-term may cause some sort of damage and/or failure.

I think my best option may be to simply remove the barrel adjuster so it doesn't abrade the cable over time. Is there any danger/drawback to doing so? (As far I am aware, the main role of the barrel adjuster is just to fine-tune the tightness of the brake without tools being involved. I can live without this convenience.)
I have to say this first: riding a bike when part of the brake system is not assembled as it is supposed to be is potentially dangerous.

The brakes may seem to work fine but if the housing or adjusters are not seated properly they may come adrift and leave you with a non functioning brake. 

Brake and shifter barrel adjusters are necessary to fine tune caliper engagement and shift indexing. I do not recommend removing them as one day you will wish they were there.

Backing a barrel adjuster all the way out will not strip the threads as there is no thread engagement therefore no damage to the threads. They can be a pain to get back in though. If you disengage the cable at the caliper temporarily so there is no tension in the cable it may be easier to screw them back in. Alternatively have a local bike repair shop sort it out.

You may also want to check out some online resources on proper brake adjustment. Park tool has a particularly good set of videos on their YouTube channel.