We probably don't see more because they are not native to Great Britain.
It nests on mountain peaks and is found in Egypt, India, and China.
Since the story is from Harry's POV and he has never been to those places, it seems reasonable the only one we see is Fawkes.
Keep in mind as well they are very hard to domesticate
The phoenix gains a XXXX rating not because it is aggressive but because very few wizards have ever succeeded in domesticating it.
and they may be difficult to track down to begin with as
... it can disappear and reappear at will.
All quotes from Fantastic Beasts and Where to Find Them - Phoenix
In addition the incubation/gestation period may last several years, which might be a good reason for overall low populations to begin with.
Phoenix eggs are glossy green or blue in colour, they require no incubation but may not hatch for several years.
Phoenix artwork from Harry Potter and the Chamber of Secrets by Jim Kay
However, the "canonicity" of this information is unknown as it comes from a secondary source.
Moreover, we do not actually know if they are truly immortal. Fantastic Beasts does not use this phrasing
The phoenix lives to an immense age as it can regenerate...
and Pottermore does not list as one the their "magical abilities"
Ability to burst into flame when their bodies begin to fail, then be reborn from the ashes. Can carry incredibly heavy loads, tears have healing properties