DH and I have always worked FT, no family support nearby. When DH was spending 50% of his time on a project the far side of the globe, we had an au pair (3.5 years) and after that, we had a lady who came for an hour in the mornings to get DD to school (so we could get out on time for our commute) for another couple of years. And afterschool club to cover the gap between ECAs in school and me getting home.
Appointments were a mixture of DH and I (mostly I but Dh did his share) taking annual or flexi leave. Illness was a case of juggling whose diary looked worse on day 1, then managing diaries so we each had either morning or afternoon in the office and both caught up at home while minding DD on the alternate slot or late into the evening, and depending on how ill she was, either driving in to the office of the person who had done the morning shift for handover and car swop (preferable if DD was well enough for car ride) or the morning person coming home to let afternoon person go in.
Afterschool activities in primary were either in school premises in the afternoons (and there was an afterschool club for childcare after that), or at weekends. Our PTA was very good so there was a wide range of sports and other activities to choose from. Swimming lessons were in the late afternoon and I was mostly able to juggle getting her there (often sitting in the gallery reading through reports for work while other parents had a great time chatting).
Weekends were for GAA, hockey, Cub Scouts, athletics, coder dojo, sailing, soccer etc. (We didn't do all of those at once, and some we never got to join, but DD wanted to do all these - and did a fair few over the years). There were also music and dance and drama groups operating at weekends if she had been interested.
She has some SNs which meant additional appointments in a clinic, usually during school hours, 40 minutes from school. Which meant me commuting an hour (40 minutes on a REALLY good day) to school, at least 10 minutes collecting DD, drive to appointment, wait, have appointment, stop en route back as DD always missed lunch so needed both food and an enticement to continue to attend so we had a coffee shop stop for those days, back to school, and then back to the office. If I was lucky, my boss accepted the odd hours as a half-day's annual leave, but often he would make me take a full day but after the fact (and after I had gone back in and done a long day to make up for some of the hours I'd missed).
Dentist was arranged for weekends for both DD and myself (and now evenings as he no longer does weekends - we meet him on the side of the hockey pitch now! [GRIN]). Doctor appointments we tried to arrange for either end of the day as much as possible - either 8.30am or 5/5.30pm. (My Dr is very near my office so I can go in lunchbreak). The optician is also open on Saturdays so we might have to book a few weeks ahead but can get an appointment then.
I do also try and organise, as far as possible, appointments etc for school holidays. If I have any control over them.
Also, some bosses have been awful and some have been very understanding over the years. Officially, I should have had flexi time for years but some bosses ignored that. And some bosses have been understanding that once the work got done, that was the important thing - so if I needed to WFH some days, or take a couple of hours during the day but finish a report at home on the laptop at night - they were happy to allow that. But I've always worked in very busy areas where we had deadlines to meet far too frequently and often at very short notice.
You just got on and managed as best you could.