Earlier quoted context omitted.
One thing they mentioned that kinda sends it home to me is that the telescope is aimed at the dark and away from the sun, and the other side is especially reflective[0]. In other words, one side is too dark to see, and the other is too bright to see. [0]: https://twitter.com/NASAWebb/status/1479161991252131841
It seems to me that that isn't quite right. "Dark" just means no visible light, but an infrared or thermal camera could see stuff. The article above mentions this possibility and the problems, which are harnessing and operating at cryogenic temperatures.
Whether the telescope is reflective enough to get a good photo is another matter - I would guess not, it's designed to minimise stray light.
As for IR, the telescope is probably bright compared to the background at least for now. The main issue though is whether you'd need to actively cool the engineering camera. Cooling stuff in space is difficult because you can only really dump heat via radiation. It's probably not worth the weight to carry a separate cryocooler just for that. The inability to conduct heat is partly why it takes so long to cool down, aside from minimising thermal stress on the infrastructure.
[1] The main reason they exist is so that you have a point of reference when walking about outside, particularly on foggy or cloud days. The meteo folks also use them as visibility markers in winter - e.g. IceCube is something like 1km from the station by line-of-sight. Turning them off for a brief period is usually fine and we'd notify the station before we did it. On a clear night though you don't need them at all.