The main reason: it's hard to get there. People often think of nuclear testing as just dropping a nuke. In reality they were hugely labor- and resource-intensive operations. It took weeks to set up a test area, figure out all of the experiments and observations they were going to make, and set off the weapon. Remote sites make that difficult enough as it was (this is why the US ended up developing a domestic test site in Nevada); the logistics of testing in the Marshall Islands were tough enough. (At least one American weapons scientist, Mark M. Mills, died from a helicopter crash while participating in a test series in the Marshall Islands.)

So that already would have made it a tough place to test nuclear weapons — it's inaccessible, and you need accessibility for the test series.

Separately, the Antarctic Treaty of 1959 explicitly prohibits it in Article V: "Any nuclear explosions in Antarctica and the disposal there of radioactive waste material shall be prohibited." I only note this separately since it didn't go into force into 1961, and atmospheric testing ended for most nations with the Partial Test Ban Treaty in 1963, more or less removing the need for distant test sites.