So we have a similar family set up
OP except that I am the high earning one with a city job and DH is the low earner (a tenth of my salary). Kids are 5 and 2.5.
Our set up is interesting because it’s contrary to gender norms and as a result very different from yours.
The expectation is that I will spend time with the kids (even though I work full time) and that DH will work (even though he earns a fraction of what I do). And when those are the expectations actually a lot of the issues you’ve mentioned fall away.
DH works part time and does more school pick ups, but I do all drop offs except when travelling.
I am senior enough that I can tell work I will log off at 6pm to do bed and bath routine and then log back in later. Obviously sometimes not possible but for the majority of time it is.
we have a cleaner and otherwise share household tasks. I do more of the mental load kid stuff, partly because a lot of it can be done on my commute (medical appointments, scheduling play dates, ordering new clothes); DH does the stuff you need to be physically at home for, like laundry.
You say you are rushed and hassled etc but there will be things that you are doing that could be done by your DH, if he was really motivated to do them (as I am).
We also have a nanny. I think you’ve written this off on the basis that they might be ill, but most people only have a few sick days a year, in which case you’re much better off than sending a kid to nursery where there will be many more sick days than that. A nanny will look after a sick child in their own home and it eliminates pick up and drop off. It’s more expensive but would still cost you less than you giving up work altogether.
I think you said you’re pregnant so you don’t know yet what life is like with 2 DC but it’s absolutely possible for you to continue working and for all of you to be happy. If you really wanted to you could make it work, but the question is whether you want to.