Another possibility, although I haven't personally tried it, is to use both a whole-milk mozzarella cheese and a relatively thin layer of low-moisture, part-skim mozzarella cheese over the whole-milk mozzarella cheese. The part-skim mozzarella tends to brown faster than the whole-milk mozzarella because it contains less milk fat. If this combo works, the part-skim mozzarella might brown enough to suit your tastes while leaving the whole-milk mozzarella below it soft and stretchy. For some reason, I have found that the lower-priced low-moisture part-skim mozzarella cheeses sold under house brand names in the supermarkets tend to brown faster than other, better known brands.
You might also put part of the mozzarella cheese (whole or part-skim) on the skin before putting on the sauce, and put the remaining mozzarella cheese on top of the pizza after all or some of the toppings have been added. With a sufficient bake time, the top cheese might turn brown while leaving the cheese that was put directly on the skin soft and stretchy and create the proper mouthfeel you are after. You don't want to put an excessive amount of the cheese on the skin since that might lead to a doughy crust.
I have also found that cheeses brown faster when I move a pizza from a lower oven position to a higher oven position toward the end of the bake. Using a broiler will also brown the cheese quite quickly. However, the top heat can also adversely affect toppings like pepperoni that are typically put on top of the cheese.
I suspect that there are also different cheese blends that might have an overall browning profile that meets your personal preferences while remaining soft and stretchy.