NATO stands for the "North Atlantic Treaty Organization" and it consists of countries that are generally in the same latitude as the USA and around the broader area of the north Atlantic ocean. Did the geographical element in the name of the alliance (North Atlantic) ever created an issue, when countries that were allied to the USA but did not meet the geographical criteria wanted to join NATO? For example, Australia, Brazil or Japan. If that is the case, was there ever an attempt to change the name and rebrand the alliance into a more global military organization?
An earlier answer I wrote about NATO, Portugal and Goa might be of interest.
Which is to say: it's not just a name. The Treaty's Article 6 is pretty explicit that the treaty applies to parties and their assets in North America, Europe, the Mediterranean, and the North Atlantic north of the Tropic of Cancer. As such, even under these conditions it doesn't apply to, say, Hawaii or French Guiana, despite those being integral parts of member states (it technically probably doesn't apply to Ceuta or Melilla although Spain says it does).
In the case of the countries mentioned, the US already had mutual defense treaties with them in a broader global alliance system. Australia, New Zealand and the US signed the ANZUS Pact in 1951 (New Zealand was suspended in 1986). Brazil and all the other Latin American states plus the US signed the Inter-American Treaty of Reciprocal Assistance or Rio Pact in 1947 (since then Cuba, Mexico, Ecuador, Bolivia and Nicaragua have left). Japan and the US signed the Treaty of Mutual Cooperation and Security between the United States and Japan in 1960, which effectively commits both parties to mutual defense. This isn't even getting into other defunct US-led treaty organizations like CENTO (Central Treaty Organization) and SEATO (South East Asian Treaty Organization).
Which is to say Western-leaning, non-European countries had plenty of opportunities to pursue mutual defense treaties with the US outside of NATO, with the added advantage that such treaties didn't bind them to defense of the European continent from a Soviet attack.