I think that's only your perception tbh. I've never felt particularly "respected" because I always worked full-time. I don't recall ever particularly thinking about it at all! It's not something that ever has mattered to me.
I really can't comprehend how it's harder to be at home. I've heard it said that you're supposed to be a mum like you don't work, and work like you don't have children. I found weekends, days off, leave, so much less stressful and more relaxed. Much less pressure. There are so many competing demands that are more easily managed if you have the space to do so.
People will say, "oh the house doesn't get so dirty when you're not there all day". Well I don't know about you, but my dust has never got that memo! The bathrooms are used sufficiently that they need to be cleaned just as often. My windows never got less dirty because we were out of the house part of the day. Clothes needed washed and dried. Beds needing changed. Coming back in the evening to face the breakfast dishes that you could get washed during the day. The commute is a lot easier at home too!
As for going to the toilet in peace, I think I preferred my children coming with me to perching praying not to fart because Sheila from Accounts went into the stall beside yours.
Or waiting until everyone has left because you've done a smelly poo and you don't want anyone to know that it was you. Yeah, relaxing. Not.
And then there's family admin/responsibilities. Nobody ever considers that you might work outside the home! You ring the GP, and they're calling you back. By the time they call, you're sitting in a room with 30 other people, and no, your piles are personal to you. Or you're in a meeting when they call. The wohm are in your house in private to take the call, or can step out to take it.
You get home from work at 6pm to a note saying can little Jane please bring in a pair of brown tights tomorrow because she's a robin in the Christmas play. Jane does not have brown tights. Whereas the sahm would have known about this by 2pm and could have rustled up a pair from a friend, or nipped into a shop.
You've done your second school drop off and you're about to drive to work when you get a call to say Johnny has forgot his guitar again so can you bring it? You do, and then you're driving in traffic in a cold sweat in case you're very late to work. (I did learn to tell Johnny to fuck off in the end, it was too stressful!!)
Why is it so hard to be at home when that is where you want to be?!