There is a reason why not much people has Phoenix as pets. Phoenixes can just escape from any cage. Even if <PERSON> was able to catch one, it would just disappear in a blink of an eye.
Only way for <PERSON> to catch a phoenix would be to use Killing curse on a phoenix unaware and take it when it is a newborn chick. But again, they will grow in days and would have left <PERSON> the instance it can disappear.
Phoenix song will also strike fear into the hearts of the evil(The reason for the name of Order of the "Phoenix"). So I am guessing, Phoenix has some way of detecting evil around it as well.
EDIT: Adding this from the "Fantastic Beasts and Where to Find them" book:
Phoenix M.O.M. Classification: XXXX The phoenix is a magnificent, swan-sized, scarlet bird with a long golden tail, beak, and talons. It nests on mountain peaks and is found in Egypt, India, and China. The phoenix lives to an immense age as it can regenerate, bursting into flames when its body begins to fail and rising again from the ashes as a chick. The phoenix is a gentle creature that has never been known to kill and eats only herbs. 322
Like the Diricawl (see page 9), it can disappear and reappear at will. Phoenix song is magical; it is reputed to increase the courage of the pure of heart and to strike fear into the hearts of the impure. Phoenix tears have powerful healing properties.
With a Foot Note:
The phoenix gains a XXXX rating not because it is aggressive, but because very few wizards have ever succeeded in domesticating it.
So why bother going against something
1. That has a way to weaken you(Strike fear into the hearts of the impure)
2. Can fight you(Fawkes vs Basilisk remember?)
3. can escape your prison at any time even if captured(it can disappear and reappear at will)
4. highly rare to find(From the Foot Note)
5. Possibly detect you and fly away or bring help(Guessing this as <PERSON> songs touch the souls and <PERSON> was able to reach <PERSON> based on his unwavered loyalty to <PERSON>. If <PERSON> can detect a positive quality in a person from a distance, it might as well detect a negative quality)
NOTE: Point 2) has not happened when <PERSON> went for Unicorn blood. But since <PERSON> is known to have gathered lot of knowledge through books, he should have read the "Fantastic Beasts and Where to Find Them" as well(it was first published in 1927). 773