I agree you're not taking enough time for yourself. In the effort to rush home and spend more time with your children, you aren't calming down enough or making enough of a break - you are rushing from one stress-filled segment of the day into the next. So the "quality time" (an over-used phrase!) with your children is diminished. You hold in your frustrations with your job because you have to, and you let it out at home because you think that's safer. But it's bugging you too.
It's hard to ask yourself, "Will this matter tomorrow?" or "Is this a battle I really want to fight?" But you have to do that. A lot of times we do things in our role as parents that were done to us as kids, and a lot of times we do the exact opposite. When I was a kid, nothing I did was good enough. A 98 on a test? "What happened to the other 2 points?" (Imagine my shock when I graduated from college and no employer ever asked for my grades!) "You got into the National Honor Society? You'd better get your homework done because it looks like we'll have to out tonight to the stupid induction ceremony." (You'd think, if there was one small group of kids who didn't have to be nagged to do their homework, it was the group in the honor society!) I was yelled at all the time, spanked frequently, and ridiculed for not being as outstanding as my mother.
I swore I would never do that to my child, but still, I lapsed into it because it was so ingrained. Exactly what you're worried about. Over time, I realized I didn't want my child to think of me the way I thought of my mother. I realized the problem was my mother's shortcomings and demons, her feelings of inadequacy, that made her behave this way toward me. You are saying that your own frustrations are causing the problem, and they have nothing to do with your 3 year old. So maybe you can relate. It's an eye-opener - it was for me.
So, can you expect less? Not just of your 3 year old, but of yourself? Can you cut in half the number of things you feel you MUST do when you pick them up? If you are racing home to get dinner and do laundry, can you make 4 dinners on Sunday that you can re-heat during the week? Rachael Ray and many others have menus and simple ideas of how to do this efficiently, while turning that Sunday meal prep into a family bonding activity. That way, when you get the kids home, you can play and just spend time together, which is what they need and want!
Can you turn other things into a game with your 3 year old? For example, sorting laundry can be an exercise in learning colors. Matching clean socks out of the dryer can be a game too. Okay, you'll have to spread them all out on the couch, and it will take twice as long, but so what? It's togetherness and it's also a learning activity for her. If you have a small basket for her own stuff, she can take her things to her room and help put them away.
Can you turn setting the table into an activity for her? If you draw an outline of silverware or create a placemat showing the proper location for each utensil, plate, napkin and cup (and then maybe run them off on different colored card stock which you have laminated), she can choose a placemat for herself and for you, put some dishes in the right location, and then all you have to do is serve and wipe down the placemat with a damp sponge. It takes some prep time but it's worth it.
Can you sometimes have "breakfast for dinner" or "upside down day" so that eggs and toast is the dinner you have? Can you put a big old sheet down on the floor for a "dinner picnic" so you don't even have to set the table? Maybe use disposable dinnerware just once every 10 days so there's a day you have no dishes. I'm a huge recycler so once in a while I don't feel guilty for using disposables.
If you work in the schools, maybe you can use the upcoming vacation to really plan out some changes. Get a book from the public library on how to wind down a little, how to get the whole family involved in family maintenance activities so it's not all on Mean Ol' Mom to be the daily police officer.
It helps to know that kids don't know what "clean your room" or "tidy up the family room" or "clear the table" really mean. They can't handle big complicated jobs like that - they can do 2 things at most ("Put your coat on the hook and take off your shoes") so don't add a third!
These years will be gone before you know it, and you will be wishing your children spent more time at home with you before rushing off with their friends to the mall or their high school group to the big game, and then there's college or the working world. Take time to really relish in what it means to be a child and to see the world through their eyes.