Could you link to where you're seeing the debate? Unfortunately, Tiwanaku is often discussed in psuedoarchaeological, "ancient aliens" type circles (credit to u/CommodoreCoco and u/Bem-ti-vi), so it can be sometimes be hard to distinguish between that and legitimate scholarship.

On the subject of the "68% probability", it seems to come from a misunderstanding or misrepresentation of the data on wikipedia. In reality, it's just another way of expressing "within one standard deviation". Two standard deviations would be ~95%. All samples have error ranges - that's why its important to have multiple samples (see a bolded point below). Also, are you looking for the founding date of the site, or the date of Tiwanaku's "high point"? While the archaeological consensus for dating Tiwanaku's expansion is around 600 AD, a founding date in the late 1st or early second century AD seems to be the consensus view in several recent publications related to the site:

Marsh, E., Roddick, A., Bruno, M., Smith, S., Janusek, J., & Hastorf, C. (2019). Temporal Inflection Points in Decorated Pottery: A Bayesian Refinement of the Late Formative Chronology in the Southern Lake Titicaca Basin, Bolivia. Latin American Antiquity, 30(4), 798-817.

Toward the end of the Initial Late Formative, in the first century AD, Lukurmata was founded, and residents used and discarded Thin Redware, the only decorated pottery associated with this phase at any site. Next, there was a surge of construction activity and residential mobility in the late first and early second century AD. Major Late Formative sites—Tiwanaku, Lukurmata, Khonkho Wankane, and Kumi Kipa—were founded, and at previously inhabited Sonaji and Kala Uyuni, people built new civic spaces. (emphasis added)

Marsh, E. (2012). A Bayesian Re-Assessment of the Earliest Radiocarbon Dates from Tiwanaku, Bolivia. Radiocarbon, 54(2), 203-218.

The date of Tiwanaku’s founding is estimated using the starting boundaries of models from Bennett’s pit and the Kalasasaya, AD 90 (80 BC–AD 340) and AD 90 (30–170), respectively. In both cases, the earliest cultural material overlies sterile soil. These 2 data sets are independent and strongly agree, despite large error ranges. Assuming both models share a starting boundary provides a composite idea of Tiwanaku’s founding date, AD 110 (50–170). This result takes into account data from both excavations, and seems to provide the most reliable estimate for the founding of Tiwanaku, given the limitations of the data. More recent research has confirmed that there is no clear material evidence at Tiwanaku that predates the Late Formative, despite its long history of research (Marsh, forthcoming). Currently, the only other Late Formative 14C dates at Tiwanaku come from Kk’araña, a residential area north of the Kalasasaya. The earliest occupation, lying over sterile soil, dates to AD 220 (130–320) (Marsh 2012: Table 7.2). (emphasis added)

In this article, Marsh also elaborates on some issues with previous dates. Apologies for the length of the quote and for leaving the parenthetical citations in, but I think it helps make the point that refining archaeological chronologies involves a lot of cross referencing.

Ponce Sanginés [1981] suggested that Tiwanaku was founded around 1580 BC, the median of the earliest date (GaK-194). The first phase, Tiwanaku I, was dated to 237 BC (Ponce Sanginés 1993:65). This is an arithmetic average of the means of the 9 earliest dates, a poor means of assessing the absolute date of the associated archaeological material. Stylistic comparisons of Tiwanaku’s earliest ceramics and monoliths suggested that the site could not be much older than a few centuries BC (Rowe 1963: 8; Chávez and Mohr Chávez 1975:66). The earliest date was excluded as an outlier decades ago; current consensus is that Tiwanaku was founded no earlier than 300 BC, slightly before the beginning of the Late Formative period (Kigoshi and Endo 1963:116; Browman 1980:114; Bermann 1990:86, 1994:66; Mathews 1992:66–7; Janusek 2003:47; Stanish 2003:117). Late Formative ceramics are the earliest diagnostic ceramics at Tiwanaku (Marsh, forthcoming). (emphasis added)

So, Marsh argued for pushing the date of the site's founding back from 300 AD to 110 AD in 2012, but I'm not sure if that's the disagreement you're referring to.