kinda hate to say this, but we’ve been doing just fine without any of them, for over 10 years now

We’ve never had a dishwasher, our gas dryer didn’t have a hookup when we moved into this house in 2000 and we’ve never bothered to install one, and our range died around 2005. We hang our clothes on a line or rack to dry (inside when it’s cold outside, but we’re coming up on the dry-stuff-outside season). We have a big toaster oven and a hotplate which both sit on our countertop and serve whatever cooking tasks we have, and we have a dish drying rack next to the sink. It’s really not the end of the world.
Having said that, we don’t have little kids; that would probably change the math a lot for us because our meals and laundry loads would all be bigger. But for just the two of us, it’s fine. I remember from being a kid that washing/drying dishes was part of the teenager to-do list, and we always dried laundry on the line. So even with older kids in the house, some division of labor would take care of two of those three appliances.
I’d suggest getting creative with “doing without” for awhile, and save up the money for general repairs rather than dipping into the emergency fund. You might find, as we have, that we really don’t miss those appliances. Or, you might find that you really truly gotta have one of them repaired NOW, but the other two can wait. And then you’ll have your answer. And your money.
Also, thanks Eldred for sharing that link. We’re firm believers in DIY work, and we repair most of our own stuff these days. That can save 90% of the repair bill, if/when you decide that appliance really does need to be up and running.