I know you directed your question to nothingisblackand white but I have the same experience - it was a combination of schedule, living space and not trying to do all the household stuff myself as well.
Schedule:
My work is based around delivering a project to a deadline but can be completely self-scheduled, all communication with clients was using online messaging sites like Slack or Basecamp so sounds in the background wasn't an issue. I could play music and sing along with her without disturbing anyone else.
I had to be very flexible so that I could jump into the complicated focused work for short periods at a time where she was napping (40 mins to 1.5 hours), or was playing quietly on her own (10-20 minutes). I saved the work that can be done with half an eye on other things to cover meltdowns or breastfeeding/feeding etc. I also used a chunk of time in the evening or morning when DH was home to work.
Luckily my work schedule was flexible enough to go to baby classes in the day, or host playdates, or take advantage of the good weather and go outside and play whenever possible. Playdates were also good if another mother wanted to get a bit of work done and we could tag team the babies while the other mother worked.
Living Space:
I used those freestanding baby gates to create a working and playing space which had my desktop computer and all the sensitive wires outside of a huge gated play area which included a sofa (before the climbing started) so that I could take my laptop in there if needed.
The space was babyproofed so I had somewhere secure where she could not hurt herself if I wasn't in there with her. I kept different baskets of toys in rotation so that she didn't get bored. And from early on I encouraged a lot of self-regulated play - 5 minutes here and there at first but she was a really busy-focused baby and at 1 years old could easily entertain herself with toys and books for 20 minutes - 30 minutes at a time. I may have just lucked out with her personality with this.
Household stuff:
You can't do all the household chores as well if you wfh with a kid because you've used up all your 'not entertaining or holding the kid time' on your work. So DH stepped up and took over stuff like laundry, I batch-cooked and bought more ready-prepared meal items. Someone else did the cleaning etc.
And it ebbed and flowed. Some periods were easier - newborn stage, crawling but pre climbing stage etc. Some periods were intense - teething etc.
You've got to be able to be flexible. Have total support from your DH. And if possible connect with Mums in similar situations and help each other out with work/playdates.