WaxOnFeckOff my approach is the same as other managers in my (generally flexible, but within limits) organisation.
A one off late start or nipping out for 30 mins during a working day as you describe, is simply a one off and would be fine.
Permanently changing your working pattern on a daily or weekly basis is rather different and would require a formal flex working request.
Your colleague who would normally start late after dropping her DC off - presumably that late start is a flexible or part time working arrangement; why would she not now have returned to that (which is what staff in our company are doing now schools have returned) rather starting early, logging out to drop DC off and then logging back on. Surely returning to the previously agreed times makes more sense?
Any role which involves telephone use (not a call centre, but anyone dealing with other 9-5 businesses) will need employees working fully or mostly between those core hours not, as one person I have heard of tried to negotiate, working a split of 6-8.30am and 4-9pm.
I appreciate the OP is only talking about morning/ lunch pickups not such a drastic change of hours. But as had been said without knowing the ins and outs of his role and his employer's approach, it's impossible to say if it could be accomodated.
For example, a drop off could enable him to start at 9. But if he currently works 8-4 rather than 9-5, his employers may well require a flex working request to change those hours on certain days to reflect his availability. Again this is something we would do - the request would be approved without difficulty, but it would be important for staffing levels and HR responsibilities that it was formally logged.