No roots showing on cutting.

Japanese maples are notoriously difficult to grow from cuttings and have a very low success rate, usually somewhere around 1-2%. So it doesn't sound promising for your broken branch.

Timing, heat, humidity and planting medium could all factor into the success rate. Rather than attempting to root in water, I might try dipping the cutting in rooting hormone, potting into a suitable growing media and keeping the start warm and under moderately humid conditions. Ideally, a late spring softwood cutting would likely give you the best results. When potted up, these often have a root system established by late summer.

But that success rate remains the same........just doesn't look too promising for a single cutting attempt.

Many popular garden plants are not grown from cuttings by commercial producers. Named forms of Japanese maples are usually grafted by them. Sometimes some kinds of plants are grown by grafting or tissue culture for particular, commercially motivated reasons but many other times the plants are just too hard (effectively impossible) to root from cuttings even when not trying to make a buck from them.