My chicken soup always ends up with bones
I generally make chicken soup in two phases. Phase 1 is chicken stock, simmering (roasted) bones with aromatics for a long time (often, overnight), followed by straining. Phase 2 is chicken soup, combining the strained stock with chicken meat, vegetables, and other ingredients.

Because the stock is strained, it has no bones, big or small. There can be vegetables in the stock as well, which will be cooked until they are mushy, giving up their flavor but no longer something you want to eat; they are also strained out. If you're using very meaty bones, or even whole birds, you can salvage the meat by picking it off the bones, and adding it to your soup (though most of the flavor will be cooked out of it.) 

The bones and the meat don't have to come from the same bird. You can take bones leftover from a roast chicken or other meal, and make stock out of that. You can even save the bones in the freezer, accumulating them from a number of projects, then make one big batch of stock. Freezing the stock means you'll have it on hand any time.

Then you could make chicken soup by buying drumsticks (or whatever part you like), and removing the meat. That will turn into soup fairly quickly: it will take just half an hour or so to cook vegetables to the right point, and just a few minutes to incorporate the meat. The bones you removed can go into the freezer, and the process continues.