I laugh when people tell me they have "closed herds" because NOT ONLY are goats houdini's and tend to get out once in a while, but unless you live in a sterile part of the world, deer and other wildlife can spread diseases like TB, brucellosis and blackleg (among other things) to your goats and cattle. I'm constantly hearing a neighbor say she has a closed herd as her animals roam the countryside instead of being where they're supposed to be.
There's no such thing as a closed herd. If you want disease free animals, the best thing to do is to have them tested. If you're going to be drinking the milk, it's a must. If you're not worried about spreading stuff to other goats or cattle, then I guess it'd be okay. They're animals, they get diseases. TB and brucellosis are human transmittable (if you drink raw unpasteurized milk). Blackleg can kill them though. CAE is hard to diagnose because it's slow-going. CL can show up later too.
I'd much prefer to do all the testing, but it really depends on what you want these animals for. Nothing is fool proof, you just do what you can to give yourself the best chances for a disease free herd. Keep in mind that if the goats are bred, or just kidded, their tests might come out false negative/positive.
For example, some diseases are slow developing, and some don't show symptoms at all. Like BLV in cows, which is basically a wasting disease. It's hard to find a cow from a dairy that doesn't test positive for it, but symptoms may never show up ever, but if it does, the poor thing wastes away to nothing. It's also transmitable to other cows. But if you already have a cow that tests positive for BLV, you might not mind getting another one with it because not only is there a small chance they'll show the devestating symptoms, but you already have a cow positive for it...in addition, it's not transmittable to humans. So it depends on your own needs and situation.
A goat with CL may not show symptoms until they spread it all over your pasture and shelters, and sometimes you might buy a kid from a doe that doesn't show symptoms until years later. Some people don't mind CAE goats if they bottle feed their kids.
But whatever you decide to do, I wouldn't just go by them saying they have a closed herd. That's just them being silly... there's no way she would possibly know they were negative by just saying she's got a closed herd and only brings in bucks that test negative.
Does she even vaccinate them? For what? And when? Be sure to ask her if they'll need boosters.
Though I think it's great that's she's providing you with the feed, grain and minerals she's providing. What about wormers? When were they wormed last, and with what? That might be helpful information for you as well.
They're very pretty though. As for the head being too big, I see what you mean, but I bet it's just the picture. You can ask them to send you more pictures...that shouldn't be a problem.