Newt's situation was unique - and he had a talent with creatures that most wizards don't
Newt is a magizoologist, who had no intention or expectation of facing off against a Dark wizard. He had his suitcase of magical creatures with him when he found himself suddenly in the position of confronting a villain, and he simply improvised and worked with what he knew. This was a situation unique to Newt. Many magical creatures can be difficult or nearly impossible to work with. Newt, however, has a way of connecting with his creatures, which is probably the reason he could get one to help him. So far, we haven't seen any other wizards with a similar ability. Also, he uses his creatures to get himself out of the situation because he wasn’t a typical ‘hero’ with battle skills.
“But Newt’s got complete control of it,’ says Manz, and this was one more consideration. Their wizard wasn’t an action hero – he uses the beasts to get him out of a scrape.”
*- Inside the Magic: The Making of Fantastic Beasts and Where to Find Them *
Certain aspects of wizardry are easy enough for most wizards to learn, like Transfiguration and Charms. While special talent is these areas is still possible, almost every wizard is expected to have at least a basic knowledge of them. These are what make up the Hogwarts curriculum. Defense and offense spells like the type used in duels, while they do take practice to master, seem to be something that the majority of wizards can master if they put their minds to it.
However, there are certain specialized skills which a smaller number of wizards have any possibility of being competent in. They may be ones that a wizard is born with, such as being a Metamorphmagus, or they may be theoretically learnable, like being an Animagus or a Legilimens. Being a Metamorphmagus, an Animagus, or a Legilimens would likely be useful in battle. However, while they would likely be valued, none of these skills are actually required to become an Auror or a Death Eater, or even particularly common among their members.
Newt's skills handling magical creatures seems to be a special talent, and a quite rare one as well. He's probably not the only one who can do it, but he's certainly one of a relatively small number. Even Hagrid, the only other person we know of that has a similar love of even the most dangerous magical creatures, had a much harder time trying to control the creatures under his care.
While handling magical creatures might be useful in battle, it's not easily picked up, and therefore it would probably not work well attempting to use creatures in battle. Trying to teach everyone to handle creatures like Newt would still likely not have them reach the same level of proficiency that he has. It would be like trying to teach Legilimency to people without any aptitude for it. Even the best among them would constantly having to work at reading minds, and Queenie Goldstein is because of her natural skill effortlessly reading minds and having to work at not doing it.
Also, the risk of handling creatures and using them in battle would likely outweigh the potential rewards. If a wizard overestimates their skill handling magical creatures, things could go very wrong, very quickly. Even if they do everything right, there is still too much risk of something going wrong. The creatures might not want to harm their owners or anyone else they are used to, but they probably wouldn't be able to recognize everyone on their owner's side of the war, and may end up attacking people on the side they're supposed to be fighting for. It's an unwise strategy to use anything in a battle that could do as much harm to your side as it might to the other side.
An additional reason why creatures are not used in battle more often: many of them, especially the kind that would be useful to have in a fight, would be protected by wizarding laws and kept in areas inaccessible to the general wizarding population. The law itself would be enough of a deterrent for most wizards, certainly for using them as part of Defense Against the Dark Arts classes and use by Aurors. For wizards who are more willing to resort to outside the law, the laws and secluded nature of the protected areas where these creatures would be kept would still make it much harder for them to obtain them, and adds the risk of them being caught for illegal creatures.
