Is there more than one good way to treat broken elbow?

Depends on fracture. For most elbow fractures, cast immobilization works well. Unfortunately, to obtain the best result, some fractures with displacement especially in the growth plate may require open reduction and fixation for the best longtime results.
Of course. Many elbow fractures are non displace and need no treatment. There are several different types of fractures of the elbow in children and if they are displaced or angulated, then surgery is necessary. Some of the surgery can be just pushing the bone into the correct alignment and then putting on a cast and sometimes a pin to hold it in place while it heals. Others require surgery to fix the bone.
Several. Depends n the specific nature, location and degree i'd displacement of the fracture. Some require surgical fixation while others need time and a splnt. Get an expert opinion quickly so all your options are available.