Although it might have been safer by some measures not to move the Palantir far, Sauron had control of it for over a thousand years. To wait until the end of the Third Age (he declared himself openly in 2951) to use it seems a little strange.

In "The Palantiri" Tolkien writes:

"Two things contributed then to the neglect of the Stones, and
their passing out of the general memory of the people. The
first was ignorance of what had happened to the Ithil-stone:
it was reasonably assumed that it was destroyed by the
defenders before Minas Ithil was captured and sacked; but
it was clearly possible that it had been seized and had
come into the possession of Sauron, and some of the wiser
and more farseeing may have considered this. It would appear
that they did so, and realized that the Stone would be of
little use to him for the damage of Gondor, unless it made
contact with another Stone that was in accord with it. It
was for this reason, it may be supposed, that the Anor-Stone,
about which all the records of the Stewards are silent until
the War of the Ring, was kept as a closely-guarded secret,
accessible only to the Ruling Stewards and never by them
used (it seems) until Denethor II."

Saruman did not take possession of Orthanc until 2759, more than 700 years after the Ithil Stone was seized. In that time, it appears no one tried to use the other Stones to contact the Ithil Stone or look at whatever region in which it might be located.

According to the Tale of Years, Saruman didn't use the Orthanc Stone until around the year 3000:

"The shadow of Mordor lengthens. Sarumans dares to use the
palantir of Orthanc, but becomes ensnared by Sauron,
who has the Ithil Stone. He becomes a traitor to the Council.
His spies report that the Shire is being closely guarded by
the Rangers."

Michael

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The Fairy Tales of J.R.R. Tolkien