Ideas
Nothing to say about solving this one, but I will say it's not necessarily "a bug", but an aspect of how activities work. essentially, Business Activities simply push a list of interactions on Sims- they don't give them a chain of commands that flow into each other. They quite literally tell the Sims "Do these interactions while you're here". Several things had new autonomous versions added for this exact reason, as they weren't doable by non-played Sims before at all.
For Laundry, the tuning looks like this:
The activity literally tells sims they are allowed to put things in washers, take things out of washers, put things in dryers, and take things out of dryers. Since you can't do the last 3 until the first is done, and has time to complete, the result is that sims show up, do the only available option of putting laundry into a washer, and then have nothing to do, and leave.
If your business has other things to do, they may stick aroudn long enough to do the other actions...but there's nothing guiding them to do their laundry, so you'll see customers take laundry from a machine they didn't even use and move it to a dryer.
This is less a bug and more a major design oversight, and it won't be "fixed"...it'll be implemented, as it's currently just simply not implemented at all. I'm not sure how this one made it to release, as it would take literally the most basic testing to realize that this isn't going to work at all.
My suggestion would be that when a Sim does the first interaction they are given a buff that encourages them to stick around a set amount of time, and when the buff disappears, pushes the interaction to move clothes to a dryer. This still wouldn't solve the issue of Sims using random washers/dryers, though, and they'd all be competing for whatever was available at all times. Essentially this needs to be its own entire new system to actually work right, so don't hold your breath, considering we're still playing the game where Thriftea basically doesn't work whatsoever.