You will only need the tool in question if you are planning on putting in new bearings.
Special tools for this would be great but I got mine out with a stout pair of needle nose pliers that would span the gap between the 2 slots to turn the retainer.
Most importantly, you will need to drill out the indentations that were punched by the factory to prevent unwanted loosening of this bearing retainer. Get a drill bit that is just a little larger and drill it just a hair deeper than the punch marks. There should be 4 of these and they are located at the outside edge of the retainer.
The retainer is made of very soft aluminum and the threads will strip if you don't drill these out but manage to turn it.
Spray a little penetrating oil around this retainer and it should come right out.
My bearings were shot more than I could have imagined.
If you are not removing the retainer and just want to separate the brake shoe housing from the rest of the wheel, this should be easy-peasy-lemon-squeezy.
After you remove the axle, it should almost fall apart. If it does not as you stated above, I would re-attach the brake lever to the rear wheel and move it back and forth a bit while trying to separate the housing from the wheel. Your shoes may be holding it in place.
I had the same problem on my wheel.
One final note. If you are putting new shoes on, before you take the old shoes off, wrap them with some fine grit sand paper and re-assemble the wheel. Put the re-assembled wheel back on the bike and rotate the wheel by hand (while on the center stand) and lightly apply the brakes. Do this several times and you will clean the inside of the drum and remove the imperfections caused by the old shoes.
When you put your new shoes on, you will have an factory-fresh like surface on the drum.
And whatever you do, keep the greasy hands off the new shoes.
Johnny