I think you and DH just need to own the grind of it. You already work 4 days which lessens pressure anyway.
Meal planning and supermarket deliveries. Just get them as and when you need. I can't always stretch to a week. This really minimises top up shops and saves money, and definitely time in the long run.
Batch cooking... I don't batch cook on a Sunday for example, but I try to cook one meal each week with a portion that will freeze. That way I'm always ahead.
Really really really sort out everything that you need every day. Lunch boxes, water bottles, coat and shoe storage, book bags. Have an easily accessible place for everything that isn't cluttered so that you can grab and go.
Make a time to empty the dishwasher and make packed lunches and stick to it. Whether it's late at night or very early morning, just do it.
As PPs have said. Deal with admin ASAP. Put dates of parties/events/deadlines in your diary straight away.
Engage with your class WhatsApp group. Go to birthday parties and meet other parents.
No TV or devices in the mornings.
A basket of final things that you keep in your downstairs loo if you have one. Wipes, toothbrushes, toothpaste, hairbrush, facecloth, sun cream etc. for the whole family so that no-one has to disappear anywhere to do anything at the last minute.
Consistent locking up routine. For example everybody and everything in the hallway together before anyone goes outside.
Coffee meeting on a Sunday morning with your DH to plan your week together.
No ironing in the mornings. Either things are pre-ironed, or not ironed.
It's hard work but this is the only solution I've found!