For the NH Stacker wagons, your understanding is correct. There are different models, though with features you might consider. I have the self unload, single bale at a time option (but don't use it). There are larger wagons with higher capacity and even some self propelled ones having a simple cab and front drive axle.
The only thing to change in your current operation is called a 1/4 turn chute on the baler which puts the bales on the ground with the cut side up (narrow side up) This is necessary to accomodate the ground pickup chute on the stacker. Not sure how you would want to integrate it into your baler having the kicker pan in place.
I can show you some pix of my deal if you are interested. Actually they are already on the TBN forum in a year old thread there somewhere in cyberspace.
If you are doing a couple of hundred or so bales a night to have fun, produce hay for your own animals, sell it to neighbors, etc. and are short of help, this system works very well. I usually drop off 2 loads or so from the stacker at my neighbor's barns for their own horses. This means I don't have to load it into my barn. They have to deal with it in their barn.
BTW, if you are having trouble with bales tumbling back down the elevator, the angle is too steep. If that's fixed by the barn story height, you can take out every other chain cleat to help keep the chain contact flat. This means learning how to break such a chain open, remove the cleat link, and put it all back together. Not impossible, but troubling for some, easy for others. There's even a special tool made to make the chain separation easier.
If your hay mow is large and long, you could consider getting a mow elevator. This is a horizontal equivalent of the lift elevator except the motor is bidirectional. It will unload the barn, too. I have dump stations in mine. That means I can control where in the mow the bales go by pulling the diverter control ropes from the ground level. These diverters toss the bales off thto the side at the diverter station in the mow. This especially handy when you get some help and the the lift elevator is packed full and the bales are coming up so fast a person in the barn (i.e. ME) can't keep up. You just dump piles at each station and deal with it later.