For this we need to reference the US Antarctic Program Inter-agency Air Operations Manual
First off this will tell us who is in charge down there (at least for US bases) as its not quite an ice frontier free for all,
Oversight of the on-ice operations of the civil aircraft is
NSF-delegated to the Dept of the Interior/OAS Office of Aviation
Services (OAS) in accordance with the SLA between OAS/AIL. Joint use
of the USAP airspace in Antarctica requires close coordination between
the DoD and NSF (and its delegations).
It looks like if you want to fly down there you need to file form DD 1801
2.3.4. Intercontinental flights between New Zealand and Antarctica will require USAP aircrew to file a DD Form 1801 (International Flight
Plan) in Christchurch Base Operations to Christchurch Clearance
Delivery. Flights utilizing the Macquarrie route will require a FIR
estimate for Melbourne and Auckland boundaries.
The document does not list any approach procedures for NZSP but it does list them for McMurdo Station and they are presumably similar,
18.104.22.168. IFR. McMurdo ATC retains responsibility for all IFR aircraft. Aircraft will be USAP Inter-agency Air Operations Manual, 1 August
2012 40 instructed to contact Tower prior to entering Class E
22.214.171.124. VFR. Aircraft are normally instructed to contact tower no closer than 10 miles from the airport for landing instructions.
126.96.36.199. Aircraft operating in Class D airspace will be in radio contact with the tower and should remain at 1,000 feet AGL or higher
(500 feet for helicopters) until commencing final descent.
They even have reg's for uncontrolled fields
2.8.1. For operations at uncontrolled McMurdo area airfields, fixed/rotary wing aircraft pilots will follow uncontrolled field
188.8.131.52. Flight plans will be filed with Mac Center and advisory service provided.
184.108.40.206. Fifteen (15) minutes prior to departure from an uncontrolled airfield, the pilot will notify Mac Center/Tower of planned activity
via landline or radio.
220.127.116.11. Mac Center/Tower will notify Aircraft Rescue and Fire Fighting (ARFF) (who notifies the USAP Airfield Manager) on FM Channel
2 of intended uncontrolled airfield operations.
18.104.22.168. Pilots will announce their intentions on the Common Traffic Advisory Frequency (CTAF) VHF 129.7 MHz.
Interestingly it looks like McMurdo is the primary landing strip however based on the letter of 1.2.1 the South Pole Station (which Im pretty sure is NZSP) only accepts ski equipped aircraft so you would at least need skis.
1.2.1. Primary airfields for the USAP are located in the McMurdo Station area and at the South Pole Station (ski-equipped aircraft
only). The primary heliport is located at McMurdo Station. Additional
ice runways, skiways, and helicopter landing sites are established
throughout the continent supporting the USAP science activities as
planned by the NSF Prime Contractor’s Fixed/ Rotary Wing Coordinators.
According to the doc a lot of the info is published in the Antarctic Flight Information Manual which looks like you may need to set up some kind of account here to download or "contact [your] affiliate National Antarctic Program".
All this in mind, I have not yet found any documentation or out right publications that seem to forbid flying there as long as procedures are followed. With that in mind you clearly need to file some flight plans and related info, I would think you would need a good reason to be going there to get approval. The reality is there may be more info on this buried in the antarctic treaty but in practice I dont know how that sort of stuff gets enforced.