The OP never mentioned venturing 50nm from original point of departure.
Correct.
And, others claimed that a 'round the world flight didn't count for any ratings.
The OP never mentioned venturing 50NM from original point of departure which again is where the intent vs the clear language of the Reg comes into play.
The OPs proposed flight is arguably one of those "questionable" ones as far as repositioning goes. Although his total distance traveled covers 140 NM, he never goes only his first stop takes him further from home than 25NM and that still is only 37NM. He does have one leg (KTOP->KMKC) that is near to a XC in its own right at 48NM but its only 22NM from his point of departure.
The question though of intent vs regs is more subtle... Here's a fictional example:
Pretend there is an airport Airport 15 NM Northeast of Homeland (HDF) VOR (4NM Southwest of KBNG) that we'll call KFIC
KCRQ to KFIC - Leg 49 NM
KFIC to KSNA - Leg 49 NM
KSNA to L08 - Leg 82 NM
L08 to KCRQ - Leg 49 NM
Only one leg is more than 50NM yet the entire flight totals 236NM and is right on the edge of XC time but unless you reposition at least once, you would be unable to log the flight as XC for rating purposes. In fact, the KSNA-L08 leg is the only section that qualifies.
KCRQ to Fictional (KFIC): 49NM
KCRQ to KSNA: 44NM
KCRQ to L08: 49NM
No Departure from KCRQ qualifies.
Fictional (KFIC) to KCRQ: 49NM
Fictional (KFIC) to KSNA: 49NM
Fictional (KFIC) to L08: 48 NM
No Departure from KFIC qualifies.
KSNA to KCRQ: 44NM
KSNA to KFIC: 49NM
KSNA to L08: 82NM
Only Departure from KSNA to L08 qualifies.
L08 to KCRQ: 49NM
L08 to KFIC: 48NM
L08 to KSNA: 82NM
Only Departure from L08 to KSNA qualifies.
It's entirely possible to fly a 49NM DME ARC with 6 stops no further than 49NM from each other and the origin airport not have it qualify as XC time because you never "repositioned" or you repositioned at each stop so no airport was more than 49NM from the last. Your total distance flown would be 392NM. I'll grant you that the case is probably completely fictional as you're unlikely to find an airport with 6 airports spaced so perfectly as to be so tantalizingly far yet still too close.
I have a hard time believing that despite the lack of an airport being 50NM from the point of origin, that the person making this flight did not acquire the aeronautical cross country experience required for a rating but the plain language regulation specifically says the entire flight would not count as XC unless the pilot repositioned once and separately logged an outbound and inbound leg of travel (in essence repositioning a 2nd time).
Which goes back to the non-stop round-the-world trip... First of all, Im not familiar with the group that did it or what their ratings were but I assume they probably have enough time logged that they do not need the time to be specifically logged as XC for ratings... Even if they did not, I imagine the FAA would make some form of special review/dispensation; after all 61.1(b)(3)(vii) allows military pilots credit for essentially doing the same thing (though they dont often circumnavigate).
For the OP: I forgot to mention a small point of order... Technically, KK81 does not exist as IATA and ICAO airport codes (airport codes starting with K in the continental US) are letter based only and neither IATA or ICAO permit numeric codes. The airport should be referenced as K81.
K81 is an FAA identifier and not having an IATA or ICAO code would be filed on your flight plan as ZZZZ for ICAO or K81 for FAA. IATA codes are used for airline ticketing and baggage routing and there are a few instances in which the IATA code can be something we as pilots would not expect.
The Continental US is the only region where FAA codes often match IATA codes which can often be prefixed with a K to get the ICAO code. In most other regions of the world, they are not issued an FAA Code and the IATA code and the ICAO code can be and often are different.
For example:
Charles de Gaulle in France
FAA: No Code, IATA:CDG and ICAO:LFPG
(L = Southern Europe, F = France, PG = Paris de Gaulle)
or
Saipen International, which is administered by the US and thus has a FAA Code, the FAA Code and ICAO code match but it has its own IATA code.
FAA: GSN, IATA: SPN, ICAO: PGSN
or a prominent continental US airport where the codes FAA/ICAO codes are the same (with a prepended K) but the IATA code doesnt match
Sawyer International in Michigan
FAA:SAW, IATA:MQT, ICAO:KSAW.
If you're going to Sawyer on a commercial flight and you see your bags tagged with SAW look out because they'll be headed to Sabiha Gockcen International in Istanbul, Turkey while you'll be in Michigan...