Each Istari had a different mission?

[Rhod the Red]

Each probably was. The Unfinished Tales chapter
on The Istari addresses these. Though it was
more, as far as I can see, the reaction or
treatment that defined the Istari.

Saruman may not have gone bad if he wasn't
grovelled so much by M-e leaders. If, for example,
(forget his name) the Steward that gave him the
key to Orthanc never did that or allowed him
access to the Minas Tirith archives, lack of access to
the Palantir & a fixed abode may have mitigated,
if not prevented, his evolution of thought.

Gandalf got fair treatment, but not 'special treatment',
as it were. So his ego wasn't tickled to inflate. Radagast
ensured he lived humbly and in 'Wild' lands on
the west side of Mirkwood. Nor did he zigzag and
seek grovelling. The Blue wizards? Who knows.

So in a way, when you think about it, the themes and
flow of the story tell us more about how they are treated
by other characters than their inherent skills. We get
hints of Saruman & Gandalf being lore-masters, but not
the full picture. And their decisions influenced by what they
saw/did in Middle-earth.