Because the case of creatures was safest in the care of <PERSON>, a trained magizoologist.
In figuring out why <PERSON> made the decision he did, there are a few things that are very important to remember.
* There was no one more capable than <PERSON> at handling magical creatures, and caring for the creatures in the case.
<PERSON>'s case contains several magical creatures, many of which are dangerous to the average wizard. From what we know of the wizarding world's attitudes toward magical creatures at the time, if <PERSON>'s statement is similar to what an average wizard thinks, they would mostly consider these creatures things to be killed or avoided, not cared for. Even if other wizards are tolerant of magical creatures, <PERSON> is likely the only wizard at the time who is actually capable of caring for this variety of creatures. There was likely no one who would be capable of taking care of the case while <PERSON> went to take <PERSON> home. So, he took the case along with him so he could continue to take care of the creatures himself.
It's perfectly logical to think the best way of handling a case of magical creatures is for it to stay with the magizoologist who's trained and capable of caring for them, and despite the problems it caused, this was still probably the best solution. Remember, the case of creatures would have needed to be taken care of for as long as it took for <PERSON> to take a steamboat to New York, return <PERSON> to the desert in Arizona, and travel back. In that time, the creatures within would likely need some kind of basic care unless the case was enchanted to feed them using magical methods. However, we see <PERSON> feeding some of his creatures, and one of Gamp's Laws states food cannot be conjured from nothing. Therefore, while this is not definitive proof because <PERSON> could possibly just like feeding his creatures even if the case could do it automatically, the case may not have any way of feeding them itself. Anyway if <PERSON> knew he wasn't intending to let the case out of his care, he'd only need such a function if he was somehow made incapable of caring for his creatures by sickness or injury.
<PERSON> also has a special skill with his creatures. 194
They seemed to mostly have some kind of affection or at least tolerance for him, getting them to cooperate with him and be more compliant. For example, <PERSON> the Bowtruckle is highly unlikely to willingly accept being out of <PERSON>'s care. They likely would be even more difficult to manage for another wizard of equal skill with creatures simply because they are used to <PERSON> as their caretaker. In addition, being a skilled wizard in general doesn't mean someone can manage the creatures. Skill with handling magical creatures is different than skill at spellcasting and. A wizard who's gifted in Transfiguration might be terrible at brewing potions, so it's similarly wrong to presume that a good enough wizard can take care of the creatures.
* The only creature who escaped while the case was actually with <PERSON> was the <PERSON>.
Most of the creatures escaped when <PERSON> no longer had the suitcase, and it was instead in the possession of <PERSON>, a Muggle who had no idea of the contents. When <PERSON> had the suitcase he was able to watch it better, and only the mischievous <PERSON> kept escaping. <PERSON> also misplaced the <PERSON> egg, which was possibly in his pocket, but the hatched <PERSON> didn't escape while the case was with <PERSON>. If the case was with someone who wasn't as experienced as <PERSON> was in handling magical creatures (whether it was in New York or left with someone somewhere else) it's possible more creatures would have escaped.
* <PERSON>'s reason for going to New York was to return a magical creature home.
He needed a way to transport <PERSON> from wherever he left from to New York, and keeping him in the case with the other creatures was a good way to safely transport him. It kept the other creatures within <PERSON>'s supervision, and also was a successful method of hiding a Thunderbird to get it past Muggle customs until <PERSON> could return him to his native home.
* <PERSON> couldn't be expected to plan for the majority of the events that took place.
The only thing <PERSON> could have possibly been reasonably expected to have a plan for is the latch breaking on his suitcase, and maybe a way to secure the <PERSON> egg better. 194