I've worked at home for 20 years, more or less, and I have a separate space (garden office), $300 conference camera and mic, huge screen, and the best internet connection possible. When this is your life you arrange things differently, and I've no doubt that others who make the change to working from home will make similar adjustments over time.
I could work with it if I was forced to do it, which I was for a few months. If I had to do it for an even longer period, a separate room would probably be necessary. But having a "black space" room is not exactly something I want in my private space either. (Not to think about my work most likely not willing to pay rent for that room :D )
I know this is a personal preference, but for me, physical distance makes my work more productive and my private time more relaxing.
Not the parent, but I use a Logitech 920s webcam. I could use one of my high-end cameras--I even have an older model Fujifilm I don't use any longer--with an HDMI to USB converter but I decided it wasn't worth the trouble and it's harder to make work in physical space anyway.
I use a Blue Snowball mic but any decent USB mic will work. I have a dedicated office and I haven't found using a headset makes it better though YMMV. I'd just as soon not wear a headset unless necessary.
Computer is just a 27" iMac with a 2nd monitor. (Though I actually usually work with video on my main screen and do any work with shared docs etc. on a MacBook.)
Also have an Elgato keylight to get the lighting a bit more even.
ADDED: Probably at least as important is that you're not backlit by a window and you have the camera at maybe a bit above eye level. I've also done a bit of "set decorating" of my background.
If I had the extra space and time to set it all up I would probably use a regular SLR camera on a tripod, and lavalier mic. But the Logitech is a lot more convenient - connects over a single USB through the docking station and just works with zero set up.
The screen is just a regular LG 4K 55" TV mounted on the wall which connects to the laptop docking station, allowing video conferences, slides, etc. to be "thrown" over there when it's convenient.
