There shouldn't be this much difficulty with the broomstick method.
Part of the problem is that so many are doing it for the first time with a fully grown rooster, they are much tougher than a young bird or a hen, there is a lot more strength in their necks and more connective tissue overall. With a bird like that, you have to place it and then pull HARD, like you mean it.
Should be laying the bird down back-up on grass or dirt, not a hard-hard surface. Rest your stick over the back of the neck near the head, use one foot on one side lightly to hold it in place, rest other foot on other side and gently pull up, to get it seated right behind the skull. Then step firmly and pull FIRMLY straight UP. As long as you're fully standing on your stick and you pull up very hard, the bird is going to pop. If you're of decent strength, the head is gonna come clean off.
Not multiple sharp tugs. Not one long pull. One soft pull to seat the stick, step HARD pull HARD, done.