WOW - This site really does attract some feministic nutcase keyboard warriors commenting, its shocking. Some here really need to step down for the day. Take a day off from the Women’s Lib movement, Jeeze. This is NOT a sexist situation.
You are in the same team here and if he is earning twice what you do, two thirds of your family income is generated by DH. Its not a case of being undermined with childcare tasks its about what is really practical here in your family unit and best use of time and resource. Financially if DH was able to take DS to school it would cost your family unit twice what it would cost if OP did it. So, its financially better for the OP to do it.
Unfortunately, when you decide to have children, school runs and all this cr@p comes with it and school times especially finishing at what can feel like just after lunch time really can take its toll on you interrupting your work schedule.
Also, why people think that WFT means he would automatically have time on his hands to do school runs is beyond me. It’s the same as running a business from home, people assume you can do domestic stuff at the drop of a hat, yet you might have calls scheduled, time commitments, online meetings scheduled you might be right up against it.
This is not to undermine the great work that the OP is doing but I think DH is being honest and upfront with her. The guy will be under pressure and if his work schedule is high, making new contacts especially in a new company then he can’t do it.
My advice would be that if OP can’t support either due to going back and having similar work pressures and commitments then recruiting help to support a nanny or similar is the answer but you need to consider the financial implications of doing that, as I say this comes back to best use of time and financial resource.
This is not some feministic argument, you should do the school run because you are female, its about best use of resource. If you were bringing in two thirds of the family income and working hours meant you were not able to schedule childcare then DH would need to fit it in, if not then you outsource.
Good luck.