When I first saw 'SP' I thought, "well what is the other parent doing?" but I've realised you mean single parent.
You are as capable as the next man, you just have double the responsibility because so few men have sole responsibility for a child.
If you are earning enough can you afford a nanny?
There is also a perception in the work place that women with children take more time off so people notice.
My mum, back in the early/mid 70s, around the time of the sex discrimination act came in got a part time job, she didn't need to financially but my brother and I were at junior school so not tiny and I think she was bored.
Anyway after a few weeks she had a conversation with her boss who said he didn't usually employ women with children for the office job because he needed someone reliable.
Anyway my mum bit her tongue and started taking not of other employees time keeping and sickness records (she had to do this as part of her job anyway I think) after 6 months she showed him how many days each of the staff had taken and the young men (it was a garage) had much worse records.
Try keeping a record of other people's time, you might be (probably will be) surprised, and it is something to show when you ask for time off.
Don't appologise, i doubt a man would if he HAD to take time for something important like a medical appointment.
OP I work in FE, all our students (I'm health and social care but usually the childcare departments are under the same umbrella) have to undertake work placements, have you thought of getting a childcare student, one on their second year of CACHE so they are almost qualified, to be 'on call'?
CACHE is the qualification you need to be a nanny or work in a nursery. I don't think doing a few pickups would be classes as 'work placement' but it might, and if it isn't then you would be looking at a 17, more probably an 18 year old who loves kids, wants to work with them and has a CRN/DBS.
I think there is a business idea in developing a sort of insurance nanny service for employers, they could offer it like a perk so if your child needs picking up from school the business could phone the nanny service and they would do the pick up. I know it wouldn't work all the time, sometimes it has to be mum or dad, but for things like the 24 hours after D and V when you have to keep them off school but they are actually fine and it's probably worth more to an employer that you can work, even if you are at home and the nanny just keeps the kids entertained.