FT teacher here. No family within 200 miles.
Looking at your examples:
collect dd1 from school once while I took dd1 to the orthodontist out of town
Well you booked the appointment, so that was entirely avoidable. All non-emergency medical appointments have to be out iof term time for us.
look after dd1 while she was off sick for 1.5 days
Childminder or I do the emergency dash to pick up sick child from school (I work closer to home than DH does). DH and I toss up who does the second day - he builds up 'flex' hiours very actively every week to have a few to 'spend' on such occasions, and he is well set up to work from home.
look after dd2 while she has INSET day
DH takes holiday, or childminder.
look after DD1 for 3 days at the start of term as her school (private) has holidays that differ from my state school term dates
I moved jobs to be in the same county, with the same holiday dates, as whemn i worked in a neighbouring county there were up to 5 weeks per year (in the worst year, 2 at Easter, 2 spearate half terms, and a week of the summer) that didn't overlap. However, childminder or holiday club type things would cover the shortdfall here.
collect both dds while I rushed off to an emergency with another relative in hospital
When at primary, both children went to a childminder before school, and after school care afterwards. So 7.30 am to 5.45 pm covered in an emergency. By 5.45, in an emergency, DH could be home (hour commute). Now DS is at secondary, and when DD starts next year, both will walk the mile + journey each way, and have had to become capable of looking after themselves before and after school. I am currently 'training' DD - 2 nights a week she walks herself home from primary, gets changed and goes out to her dance school about 10 minutes' walk away. She has so far proved herself entirely capable, so there will be more of that as Year 7 approaches.
have dd's as usual on my staff meeting nights after school
Paid after school care for DD. DS does an extra-curriocular activity at school (free) to reduce his time home alone.
Friends who are more 'there' for their children before and after school comment on how capable and independent my children are - I never have to ferry lost kit etc. I reply that they have simply had to be.